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The Non-smooth contact dynamics method
Abstract
The main features of the Non-Smooth Contact Dynamics method are presented in this paper, the use of the dynamical equation, the non-smooth modelling of unilateral contact and Coulomb's law, fully implicit algorithms to solve the dynamical frictional contact problem for systems with numerous contacting points, in particular large collections of rigid or deformable bodies. Emphasis is put on contact between deformable bodies. Illustrating numerical simulation examples are given for granular materials, deep drawing and buildings made of stone blocks.
... The model is limited to the main features of the contact, such as the non-overlapping of the grains, without further refinement. We refer to the seminal papers [18,19,31,32] for a detailed description of the so-called Non-Smooth Contact Dynamics method (NSCD). The corresponding model fall into the framework of non-smooth convex analysis. ...
... Then, the founding NSCD algorithm [18,19] choose to discretize the set of admissible velocities C c (2) using the following discrete constraint: ...
... where the expression of virtual work involves the term m(u + − u − ) · (v + + v − )/2 with u the actual velocity and v the virtual one. In (18), U stands for the pre-impact velocity, u λn the post-impact velocity, and the instantaneous contact force f is related to the momentum jump m(u + − u − ) by the fundamental principle of dynamics at the instant of contact. ...
This paper reviews the different existing Contact Dynamics schemes for the simulation of granular media, for which the discrete incremental problem is based on the resolution of convex problems. This type of discretization has the great advantage of allowing the use of standard convex optimization algorithms. In the case of frictional contacts, we consider schemes based on a convex relaxation of the constraint as well as a fixed point scheme. The model and the computations leading to the discrete problems are detailed in the case of convex, regular but not necessarily spherical particles. We prove, using basic tools of convex analysis, that the discrete optimization problem can be seen as a minimization problem of a global discrete energy for the system, in which the velocity to be considered is an average of the pre- and post-impact velocities. A numerical study on an academic test case is conducted, illustrating for the first time the convergence with order 1 in the time step of the different schemes. We also discuss the influence of the convex relaxation of the constraint on the behavior of the system. We show in particular that, although it induces numerical dilatation, it does not significantly modify the macrosopic behavior of a column collapse en 2d. The numerical tests are performed using the code SCoPI.
... Achieving this condition in a fully discrete setting with traditional contact methods is problematic. Numerous algorithms have been developed to address these challenges [44,63,58,59,19,49,42,86], though these often require complex and invasive modifications to existing methods. ...
... To mitigate these spurious oscillations, various strategies have been developed. These can be categorized into several groups: contact enforcement techniques [10,44,63,73,74,75], mass redistribution methods [49,23,38,92], stabilization methods [24,47,19], and adapted time integration schemes [30,11,16,91]. ...
... Contact enforcement techniques, commonly referred to as non-smooth contact dynamics methods, were initially developed by Jean [44] and Moreau [63] within the context of rigid body dynamics. The objective is to implement a velocity-based contact law to completely describe the impact problem. ...
Contact phenomena are essential in understanding the behavior of mechanical systems. Existing computational approaches for simulating mechanical contact often encounter numerical issues, such as inaccurate physical predictions, energy conservation errors, and unwanted oscillations. We introduce an alternative technique, rooted in the non-overlapping Schwarz alternating method, originally developed for domain decomposition. In multi-body contact scenarios, this method treats each body as a separate, non-overlapping domain and prevents interpenetration using an alternating Dirichlet-Neumann iterative process. This approach has a strong theoretical foundation, eliminates the need for contact constraints, and offers flexibility, making it well-suited for multiscale and multiphysics applications.
We conducted a numerical comparison between the Schwarz method and traditional methods like Lagrange multiplier and penalty methods, focusing on a benchmark impact problem. Our results indicate that the Schwarz alternating method surpasses traditional methods in several key areas: it provides more accurate predictions for various measurable quantities and demonstrates exceptional energy conservation capabilities. To address the issue of unwanted oscillations in contact velocities and forces, we explored various algorithms and stabilization techniques, ultimately opting for the naïve-stabilized Newmark scheme for its simplicity and effectiveness. Furthermore, we validated the efficiency of the Schwarz method in a three-dimensional impact problem, highlighting its innate capacity to accommodate different mesh topologies, time integration sc
... The popular Discrete Element Methods (DEM) [11][12][13] are based on the concept of treating granular materials as a collection of rigid particles with deformable contacts between them. As a result, in these techniques it is assumed that the particle shape and volume do not change. ...
... Then, f n ( f n = f C, pq ⋅ n pq ) are given by the following linear relation: where k n is an offset force which depends on other contact forces. The normal forces at all contact points are obtained through an iterative process by intersecting the above linear relation (12) with the Signorini graph, as shown in Fig. 2a. ...
We introduce a novel numerical method for the simulation of soft granular materials, in which the particles can undergo large strains under load without rupture. The proposed approach combines an explicit total Lagrangian formulation of Material Point Method (TLMPM) with the Contact Dynamics (CD) method. The TLMPM resolves particle bulk deformations whereas the CD treats contact interactions between soft particles. The efficiency and accuracy of this approach are illustrated by analyzing diametral compression of a soft circular particle and the compaction of an assembly of soft particles up to very high levels of packing fraction. We show that although the assembly undergoes a jamming transition, the particles continue to rearrange as they get increasingly distorted under load. Interestingly, as the packing fraction increases, a transition occurs to a regime fully governed by particle shape change. The evolution of the global stress as well as the connectivity of the particles as a function of the packing fraction are discussed and a predictive model relating stress to packing fraction beyond jamming transition is proposed.
Graphical Abstract
... As mentioned above, our simulation framework is based on the 3-D non-smooth numerical model introduced in Daviet & Bertails-Descoubes (2016b), which rewrites the plastic Drucker-Prager equations as a non-smooth root-finding problem, and leverages efficient Gauss-Seidel algorithms originally developed in the context of contact dynamics (Moreau 1994;Jean 1999) to solve for the resulting nonlinear equation. Details about this approach can be found in Daviet & Bertails-Descoubes (2016a,b), and are briefly described here for the sake of completeness. ...
... A.3. Gauss-Seidel algorithm Problem (2.15) could be solved with any technique able to address discrete Coulomb friction problems; here we follow the method of Daviet et al. (2011), which is itself a variant of the non-smooth contact dynamics (Jean 1999) algorithm. ...
In this paper, transient granular flows are examined both numerically and experimentally. Simulations are performed using the continuous three-dimensional (3-D) granular model introduced in Daviet & Bertails-Descoubes ( ACM Trans. Graph. , vol. 35, no. 4, 2016 b , p. 102), which represents the granular medium as an inelastic and dilatable continuum subject to the Drucker–Prager yield criterion in the dense regime. One notable feature of this numerical model is to resolve such a non-smooth rheology without any regularisation. We show that this non-smooth model, which relies on a constant friction coefficient, is able to reproduce with high fidelity various experimental granular collapses over inclined erodible beds, provided the friction coefficient is set to the avalanche angle – and not to the stop angle, as generally done. In order to better characterise the range of validity of the fully plastic rheology in the context of transient frictional flows, we further revisit scaling laws relating the shape of the final collapse deposit to the initial column aspect ratio, and accurately recover established power-law dependences up to aspect ratios of the order of 10. The influence of sidewall friction is then examined through experimental and simulated collapses with varying channel widths. The analysis offers a comprehensive framework for estimating the effective flow thickness in relation to the channel width, thereby challenging previously held assumptions regarding its estimation in the literature. Finally, we discuss the possibility to extend the constant coefficient model with a hysteretic model in order to refine the predictions of the early-stage dynamics of the collapse. This illustrates the potential effects of such phenomenology on transient flows, paving the way to more elaborate analysis.
... Particles repel each other via frictionless contact interactions defined by the Signorini's contact problem, and are resolved numerically via the Non-Smooth Contact Dynamics (NSCD) implicit solution algorithm (Jean 1999). Here is how this Frictionless contact is defined: ...
... For the wave speeds we considered in the study, the cell model reached mechanical equilibrium at the end of each 0.1 ms sub-iteration. Mechanical computations were achieved using LMGC90 a Non-Smooth Contact Dynamics (NSCD) mechanical solver dedicated to divided medium mechanics (Jean 1999;Dubois and Jean 2006). Implementing the motion of traveling waves and migrating cell model on a time step of 0.1 µs guarantees the convergence of the mechanical calculations at each sub-iteration and the coherence of the entire iterative process of the dynamic simulation of curvotaxis. ...
In vitro experiments have shown that cell scale curvatures influence cell migration; cells avoid convex hills and settle in concave valleys. However, it is not known whether dynamic changes in curvature can guide cell migration. This study extends a previous in-silico model to explore the effects over time of changing the substrate curvature on cell migration guidance. By simulating a dynamic surface curvature using traveling wave patterns, we investigate the influence of wave height and speed, and find that long-distance cell migration guidance can be achieved on specific wave patterns. We propose a mechanistic explanation of what we call dynamic curvotaxis and highlight those cellular features that may be involved. Our results open a new area of study for understanding cell mobility in dynamic environments, from single-cell in vitro experiments to multi-cellular in vivo mechanisms.
... Further details on the FEM-DEM framework can be found in [45]. The fluid phase is solved on a coarser scale with the volumeaveraged Navier-Stokes equations [46], while the granular phase is solved on a finer scale with the nonsmooth contact dynamics [47,48]. The two scales are coupled through an interaction force between the fluid and granular phases. ...
... where v is the volume-averaged fluid velocity, ρ f and η are the fluid density and viscosity, respectively, d is the deformation rate tensor, p is the fluid pressure, I is the identity tensor, f is the resultant of the fluid-grain interaction force tensor, and g is the acceleration due to gravity. The granular material is solved with a nonsmooth contact dynamics (NSCD) approach [47,50]. The elongated grains are modeled as rigid clusters of disks. ...
The collapse dynamics and runout of columns of elongated grains in two dimensions are numerically investigated in dry and immersed conditions, by means of an unresolved finite elements/discrete elements model. The elongated grains are modeled as rigid aggregates of disks. The column aspect ratio is systematically varied from 0.125 to 16 in order to span short and tall columns. To analyze the effect of the initial grain orientation, columns with an initial grain orientation that is either random or aligned with a given direction are both considered. Collapse dynamics, both in dry and immersed cases, are found analogous to that previously observed for circular grain columns, particularly with respect to the power law dependency for the runout as a function of the column aspect ratio. The effect of the fluid mainly results in a decrease of the runout distance. Interestingly, the collapse dynamics and runout are not significantly affected by the initial orientation of the grains, except maybe in the extreme case where the grains are all horizontally oriented, which geometrically prevents the collapse. Finally, a scaling based on the front propagation energy is proposed allowing one to unify the runout of short to tall and dry to immersed columns in a single description, regardless of the initial grain orientation.
... The method was later adopted to model masonry assemblies in which masonry blocks are modelled as either rigid or deformable. The most common method to model a deformable body, also used in the 3DEC software, is to subdivide it into finite elements, usually into uniform-strain tetrahedral elements [33,34]. ...
SiDMACIB (Structurally informed Design of Masonry Assemblages Composed of Interlocking Blocks) is the first numerical model capable of extending the equilibrium problem of limit analysis to interlocking assemblies. Adopting the concave formulation, this model can compute the stress state at the corrugated faces with orthotropic behaviour, such as their combined torsion-shear capacity. Generally speaking, finding the plastic torsion-shear capacity of planar faces shared between conventional blocks is still a fresh topic, while investigating this capacity for interlocking interfaces is particularly rather unexplored. Upon the authors’ previous works that focused on interlocking blocks with a single lock, in this paper, an extension to blocks composed of several locks (multi-lock interfaces) is presented and the SiDMACIB model is upgraded accordingly. For this purpose, the shear-torsion results obtained from the original SiDMACIB formulation are validated and subsequently compared with those derived from distinct element analysis conducted using the 3DEC 7.0 software. Based on this comparison, revisions to the SiDMACIB model are proposed, involving a reduction in the number of locks affecting torsion-shear capacity.
... To study the dynamic behaviour of stone arch bridges through the analysis of the relative displacement between stone blocks, non-smooth contact analysis is utilized. One of the most commonly used discrete element approaches for analyzing the structural behaviour of masonry arch structures is DEM [125]. DEM can use either rigid or deformable blocks and applies a soft contact approach that determines contact forces based on the joint normal and shear stiffness properties and relative displacements between blocks. ...
Masonry bridges, integral to transportation networks for over a century, have continuously adapted to evolving traffic demands and speeds, despite their original design catering to lighter loads and slower traffic. However, the prolonged exposure to diverse dynamic loads such as traffic, seismic events, wind, waves, blasts, and weathering, compounded by various environmental factors, has led to structural deterioration, presenting significant functional challenges. Identifying appropriate techniques for analysis is paramount for determining the life expectancy of these historic structures, ensuring their durability, and preserving their heritage significance. Regrettably, established protocols or codebooks for conducting assessments are lacking in many regions. The assessment of masonry bridges has evolved from initial reliance on empirical methods to the utilization of equilibrium methods, numerical analyses, and vibration-based approaches, facilitating a deeper understanding of their behaviour. Moreover, the implementation of Artificial Intelligence for assessment has gained global popularity due to its simplicity and robustness, particularly in evaluating both visual and material damage. This review paper comprehensively examines the most commonly used assessment methods, providing detailed insights into their strengths and weaknesses. Furthermore, it aims to disseminate information about recent advancements in assessment methods to both researchers and practitioners. These advancements are essential not only for laboratory-scale model testing but also for their practical implementation in real-time scenarios. Moreover, this study proposes the most appropriate method for a specific bridge considering factors such as timeline, geometry, and material characteristics. Ultimately, ensuring the long-term viability of these historic structures necessitates understanding appropriate assessment techniques and making well-informed decisions regarding conservation and strengthening measures.
... As an alternative to experimental investigations, researchers have focused more on the numerical tools (Bui and Nguyen, 2021;Lei et al., 2022;Nie et al., 2022a) to examine granular systems. As one of the most efficient numerical tools, the discrete element method (DEM), including a 'soft'-particle DEM method (Cundall and Strack, 1979) and a non-smooth contact dynamics (NSCD) method (Jean, 1999;Radjai and Richefeu, 2009), has been widely adopted to gain insights from microscopic viewpoints and to explore the relationship between particle-scale properties and macroscopic responses. For example, a well-established stress-fabric-force (SFF) framework (Rothenburg and Bathurst, 1989;Guo and Zhao, 2013) was built through DEM, which successfully relates the macroscopic shear strength of sheared granular materials to their microscopic fabric anisotropy. ...
... In addition to the non-penetration condition, the nodes in the contact area should also satisfy the conditions that the contact stress is always negative and the contact gap and normal stress cannot be zero at the same time. The constraints that need to be followed in the contact area are summarized below [39]: ...
In this paper, a new strong-form numerical method, the element differential method (EDM) is employed to solve two- and three-dimensional contact problems without friction. When using EDM, one can obtain the system of equations by directly differentiating the shape functions of Lagrange isoparametric elements for characterizing physical variables and geometry without the variational principle or any integration. Non-uniform contact discretization is used to enhance contact conditions, which avoids performing identical discretization along the contact surfaces of two contact objects. Two methods for imposing contact constraints are proposed. One method imposes Neumann boundary conditions on the contact surface, whereas the other directly applies the contact constraints as collocation equations for the nodes within the contact zone. The accuracy of the two methods is similar, but the multi-point constraints method does not increase the degrees of freedom of the system equations during the iteration process. The results of four numerical examples have verified the accuracy of the proposed method.
... For details about this method, please see Refs. [46][47][48]. We test the ESP as recently proposed in the literature and analyze whether or not it still holds once the dispersion of particle sizes is considered. ...
In this Letter, the effective stress principle (ESP)—originally developed for granular materials saturated with water and recently extended to unsaturated ones via simulations—is shown to fail once the material presents wide grain size distributions. We demonstrate that the current ESP approaches cannot capture the Mohr-Coulomb strength parameters as soon as the grain size span exceeds dmax/dmin∼4. This failure is attributed to significant differences in the fabric generated by solid interactions in the wet material, which are supposedly capable of matching the characteristics of the dry material. We show that a generalization of the ESP requires not only macroscopic considerations but also direct attention to the nature of contact and force networks.
... This method has been developed in [10,11,20,21] to study the motion of rigid mechanical particles. In this method, the contact velocity-correcting terms are not functions of the distance. ...
... (2) in Toscano Corrêa et al. (2018)) 5 that is (i) semi-discretized in space, using the finite element method, as a system of ordinary differential inclusions (eq. (5) in Toscano Corrêa et al. (2018)), and (ii) integrated in time using a special implementation of the Non-smooth Contact Dynamics Method (NSCD) developed by Moreau (1994) and Jean (1999), adapted to distributed Coulomb friction. The stiffness modulus of the foundation is k = 250 kN/m 2 and thirteen different values of the maximum force per unit length of the frictional dampers fu (0, 1, 2, 3, 3.5, 4, 5, 6, 7, 7.5, 8, 9 and and v values, the computational time to obtain the corresponding upward and downward maximum displacements ranged between 4000 and 4800 seconds, when using a computer with an Intel® Core™ i5-3470 CPU @ 3.20 GHz, 8 GB of RAM, and a 64-bit Operating System. ...
Since the use of finite element (FE) simulations for the dynamic analysis of railway beams on frictionally damped foundations are (i) very time consuming, and (ii) require advanced know-how and software that go beyond the available resources of typical civil engineering firms, this paper aims to demonstrate the potential of Artificial Neural Networks (ANN) to effectively predict the maximum displacements and the critical velocity in railway beams under moving loads. Four ANN-based models are proposed, one per load velocity range ([50, 175] ∪ [250, 300] m/s; ]175, 250[ m/s) and per displacement type (upward or downward). Each model is function of two independent variables, a frictional parameter and the load velocity. Among all models and the 663 data points used, a maximum error of 5.4 % was obtained when comparing the ANN- and FE-based solutions. Whereas the latter involves an average computing time per data point of thousands of seconds, the former does not even need a millisecond. This study was an important step towards the development of more versatile (i.e., including other types of input variables) ANN-based models for the same type of problem.
... The method is nowadays implemented in the commercial software packages UDEC (Itasca, 2022) and 3DEC (Itasca, 2023). Another DEM formulation is the Non-Smooth Contact Dynamics Method (Jean, 1995(Jean, , 1999, initially used for modelling granular flows using only spherical rigid elements, and next extended to 2D and 3D structural configurations, particularly to the modelling of masonry structures (Chetouane et al., 2005;Dubois et al., 2018;Taforel, 2012). The formulation has been implemented in the opensource DEM tool LMGC90 (UMontpellier, 2023). ...
This paper introduces a novel formulation, called Hybrid Discrete-Finite Element (HybriDFEM) method, for modeling one-directional continuous and discontinuous planar beam-like members, including nonlinear geometric and material effects. In this method, the structure is modeled as a series of distinct rigid blocks, connected to each other through contact pairs distributed along the interfaces. Each of those contact pairs are composed of two nonlinear multidirectional springs in series, which can represent either the deformation of the blocks themselves, or the deformation of their interface. Unlike the Applied Element Method, in which contact pairs are composed of one single spring, the current approach allows capturing phenomena such as sectional deformations or relative deformations between two blocks composed of different materials. This method shares similarities with the Discrete Element Methods in its ability to model contact interfaces between rigid or deformable units, but does not require a numerical time-domain integration scheme. More importantly, its formulation resembles that of the classical Finite Elements Method, allowing one to easily couple the latter with HybriDFEM. Following the presentation of its formulation, the method is benchmarked against analytical solutions selected from the literature, ranging from the linear-elastic response of a cantilever beam to the buckling and rocking response of continuous flexible columns, and rigid block stackings. One final example showcases the coupling of a HybriDFEM element with a linear beam finite element.
... Initially developed to study the dynamics of granular materials [101,157], the NSCD method is now widely used for the structural and seismic analysis of URM structures, mainly at the meso-scale. The NSCD implements Signorini's impenetrability condition, which sets unilateral constraints (unlike DEM and AEM) by ensuring that the product of the contact force (r n ) and the distance between two blocks (g) is always zero (Fig. 8) [74]. ...
In the last few decades, discontinuum (or discrete, discontinuous) numerical modelling strategies-i.e. those capable of representing the motion of multiple, intersecting discontinuities explicitly-have become increasingly popular for the structural and seismic assessment of unreinforced masonry (URM) structures. The automatic recognition of new contact points and prediction of large deformations up to complete separation are unique features of discontinuum-based models, making them particularly suitable for unit-by-unit simulations. The adaptation of discrete computational models, primarily used for analyzing rock mechanics and geomechanics problems, to the conservation, structural and earthquake engineering evaluation of URM assemblies is still ongoing, and recent advances in computer-aided technologies are accelerating significantly their adoption. Researchers have now developed fracture energy-based contact models tailored to unreinforced masonry mechanics , explored discontinuum analysis from the mortar joint-to the 3D building-level, combined discrete modelling strategies with analytical or continuum approaches, integrated the latest structural health monitoring and image-based developments into discontinuum-based analysis framework. Concurrently, new and still unsolved issues have also arisen, including the selection of appropriate damping schemes, degree of idealization and discretization strategies, identification of appropriate lab or onsite tests to infer meaningful equivalent mechanical input parameters. This paper offers to the research and industry communities an updated critical appraisal and practical guidelines on the use of discontinuum-based structural and seismic assessment strategies for URM structures, providing opportunities to uncover future key research paths. First, masonry mechanics and discontinuum-based idealization options are discussed by considering micro-, meso-and macro-scale modelling strategies. Pragmatic suggestions are provided to select appropriate input parameters essential to model masonry composite and its constituents at different scales. Then, discontinuum approaches are classified based on their formulation, focusing on the Distinct Element Method (DEM), Applied Element Method (AEM) and Non-Smooth Contact Dynamics (NSCD), and an overview of primary differences, capabilities, pros and cons are thoroughly discussed. Finally, previous discontinuum-based analyses of URM small-scale specimens, isolated planar or curved components, assemblies or complex structures are critically reviewed and compared in terms of adopted strategies and relevant outcomes. This paper presents to new and experienced analysts an in-depth summary of what modern discontinuum-based tools can provide to the structural and earthquake engineering fields, practical guidelines on implementing robust and meaningful modelling strategies at various scales, and potential future research directions.
... The simulation of granular samples with grain size dispersion is performed using the DEM known as contact dynamics (CD) (Jean, 1999;Radjai & Richefeu, 2009;Dubois et al., 2018). This method is well suited for modelling collections of rigid bodies interacting by way of frictional contacts. ...
... While some of this material is not directly relevant to the present study, a very interesting approach is to convert the complementarity formulations of unilateral contact and friction, which are governed by a system of equalities and inequalities and complementarity conditions, into an equivalent set of non-smooth equalities. These equalities are expressed as the zero-level set of certain functions which have been shown to exist for frictional contact problems [56,57], and such equivalent non-smooth equalities have been shown to lead to a simpler formulation and effective solution method for the three-dimensional frictional contact problem [58,59]. This approach, which expresses the non-smooth relationships in an implicit fashion, has also been used effectively for a simplified friction problem [60]. ...
A very compact weighted residual formulation is proposed for the construction of periodic solutions of oscillators subject to frictional occurrences. Coulomb's friction is commonly expressed as a differential inclusion which can be cast into the complementarity formalism. When targeting periodic solutions, existing algorithms rely on a procedure alternating between the frequency domain, where the dynamics is solved, and the time domain, where friction is dealt with. In contrast, the key idea of the present work is to express all governing equations including friction as equalities, which are then satisfied in a weak integral sense through a weighted residual formulation. The resulting algebraic nonlinear equations are solved numerically using an adapted trust region nonlinear solver and basic integral quadrature schemes. To increase efficiency, the Jacobian of the friction forces is calculated analytically in a piecewise linear fashion. The shape functions considered in this work are the classical Fourier functions. It is shown that periodic solutions with clear multiple sticking and sliding phases can be found with a high degree of accuracy. The equality-based formulation is shown to be effective and efficient, convergence being achieved in all cases considered with low computational cost, including for large numbers of harmonics. Importantly, this new friction formulation does not suffer from the typical limitations or hypotheses of existing frequency-time domain methods for non-smooth systems, such as regularization, penalization, or massless frictional interfaces.
... When also elastic impacts should be included, two major methods have been established. The first method, known as the Moreau-Jean method [1,[9][10][11], solves the constraints at the velocity level while incorporating a Newton-type impact law. The second method, referred to as the Schatzman-Paoli method [12,13], directly considers the constraints at the position level and is restricted to frictionless unilateral constraints. ...
In 1983, Andersen proposed the RATTLE algorithm as an extension of the SHAKE algorithm. The RATTLE algorithm is a well-established method for simulating mechanical systems with perfect bilateral constraints. This paper further extends RATTLE for simulating nonsmooth mechanical systems with frictional unilateral constraints (i.e. frictional contact). With that, it satisfies the need for higher-order integration methods within the framework of nonsmooth contact dynamics in phases where the contact status does not change (i.e. no collisions/constant
sliding states). In particular, the proposed method can simulate impact-free motions, such as persistent frictional contact, with second-order accurate positions and velocities and prohibits penetration by unilateral constraints on position level.
... Recently, the effectiveness of nonsmooth dynamics approaches has been demonstrated in many-body dynamics problems such as simulation of sand particles, stacking of bricks, and stacking of multiple balls in a box [55][56][57]; in all these applications the formulation of constraints is straightforward. When implementing a nonsmooth dynamics approach to multibody systems, a crucial aspect is the formulation of constraints. ...
Real-world multibody systems do not have ideal joints; most joints have some clearance. The clearance allows the connected bodies to undergo a misalignment, and the resulting dynamics is governed by the contacts thus formed. Two approaches are typically taken to deal with contacts: the commonly used continuous dynamics approaches assume the Hertzian nature of the contact modeled by nonlinear unilateral spring-damper elements; while the nonsmooth dynamics approach results in a complementarity problem. This paper employs a nonsmooth dynamics approach to develop a coherent framework for the simulation of multibody systems having frictionless joints with clearances. Because clearances are of small magnitude relative to the dimensions of the mechanical components, the nature of the contact in the joints is assumed to be inelastic. Using this assumption and the general nonsmooth dynamics framework, the parametric formulations for cylindrical, prismatic, and revolute joints with clearances are derived. The equations of motion are formulated, and their time-discretized counterparts are cast as a nonlinear programming problem. The proposed scheme also enforces normalization constraint on Euler parameters in contrast to state-of-the-art methods that is conducive to stability of the solution for a suitable range of step sizes. In addition, a variable time-stepping scheme that includes the step size as an extra variable in the optimization is introduced and its stability properties are discussed. The versatility of the proposed framework is demonstrated through numerical experiments.
... In the NSCD method [9,10], the rigid blocks of the structure are undergoing shock laws and Coulomb friction. The tower exhibited a complex dynamic behaviour, because of the geometrical nonlinearity and the non-smooth nature of the contact laws, with a focus on the possible non-smooth nature of the dynamic response, which usually occurs right before or during the collapse with velocity discontinuities. ...
The present paper investigates, from an advanced numerical point of view, the progressive damage of the Amatrice (Rieti, Italy) civic clock tower, after a long sequence of strong earthquakes that struck Central Italy in 2016. Two advanced numerical models are here used to provide an insight into the modalities of progressive damage and the behaviour of the structure under strong dynamic excitations, namely a Discrete Element Method (DEM), the Non-Smooth Contact Dynamics (NSCD) method, and a FE Concrete Damage Plasticity (CDP) model. In both cases, a full 3D detailed discretization is adopted. From the numerical results, the role played by both the actual geometries and the insufficient resistance of the constituent materials are envisaged, showing a good match with actual crack patterns observed after the seismic sequence.
... As well-known, the discrete element method is a numerical approach that treats materials as a collection of discrete particles interacting with Structural Dynamics Approaches for Mitigating Seismic Risk of Italian Existing Buildings (2022L-02) ... each other through contact forces. DEM has been extensively used for the analysis of granular materials, and its application to masonry structures has gained significant attention [25][26][27][28][29][30][31][32][33]. ...
The research project aims to gain insights into the non-smooth contact dynamics of existing masonry structures, which represent the major part of the historical architectural heritage of Italian cities and are sensibly vulnerable with respect to moderate-large earthquakes, that relatively often occur in the Italian territory, characterised by high seismicity rate. Additionally, masonry constructions should not be demolished and reconstructed due to their architectural and historical values, according to ICOMOS and ISCARSAH guidelines [1]. In the last decades, several numerical methods have been proposed in the inherent literature in order to faithfully grasp the actual behaviour of these structures, and, more importantly, to define ad hoc seismic retrofit interventions. Despite the numerous different approaches suggested, the debate within the scientific community is still alive and does not converge on a unique structural model. With the aim of providing a novel understanding of the complex mechanics of masonry structures, we enhance some discrete element methods addressed by the inherent literature, also giving raising further issues into non-smooth contact dynamics of rigid masonry blocks separated by mortar joints. In detail, more refined models are uploaded to account for some dissipative mechanism, e.g. the rocking phenomena and the contact friction between rigid blocks and several parametric analyses are implemented, in order to provide a comprehensive view on the competition of the above-mentioned effects on the strong nonlinear dynamics of masonry panels. These results can be helpful for defining novel smart engineering interfaces that replace the mortar joints at specific parts of the structures, thus improving the seismic behaviour of the whole complex. Therefore, it is believed that this research can respond to the need to adapt the existing architectural heritage in high seismicity areas like Italy with minimally invasive interventions that do not alter the building's structure, thus preserving its historical, artistic and architectural value.
... The particle position, velocity, and inter-particle contact forces are updated based on the impulse between colliding bodies. This feature enables NSCD to handle large-scale granular systems with irregular particle shapes at ease and bypasses the complex algorithms for computing contact forces of irregular shaped particles in the traditional penalty-based method [19,44]. In NSCD, the contact detection and resolution process are conducted by solving the differential variational inequality coupled with equations of motion through an iterative solver. ...
The concept of critical state has been a cornerstone of modern critical state soil mechanics. It remains inconclusive how grain crushing affects critical state behavior in crushable granular sand. A multiscale computational approach is employed in this study to simulate the shearing behavior of crushable granular sand at critical state. Grain crushing is rigorously considered by preserving the co-evolutions of grain size and shape in the simulation of the shearing process. Systematic simulations on specimens with varying initial states and loading paths show unique characteristics of critical state for crushable granular sand in terms of critical state stress ratio, void ratio, breakage index, and shape descriptors which are independent of stress path and initial conditions. To further understand the deformation mechanisms of crushable sand at critical state, the volumetric strain is decomposed into three components due respectively to grain size reduction, the interlocking of irregular shaped grains generated by crushing, and inter-particle friction. Competing mechanisms among the three strain components are quantitatively analyzed and discussed. Initial void ratio and stress levels are found to play a prominent role in shaping the critical state deformation of crushable sand and such impact may be gauged through the fraction of grains that experience crushing.
... The subset ∂ c Ω (i) ⊂ ∂Ω (i) is the contact surface of body Ω (i) (see Figure 1) and we let ∂ c Ω := ∂ c Ω (1) ∪ ∂ c Ω (2) . Frictional contact is modeled by combining the normal unilateral contact law and the tangential law of Coulomb's friction with variable pressure [7,8,49,50,51,30,38]. These relations depend on the normal distance d ν , the tangential contact velocity • dτ , and the contact and friction stress Γ (which is split into a normal contact pressure Γ ν and a tangential part Γ τ ). ...
In this work, we propose a mathematical model of hyper-viscoelastic problems applied to soft biological tissues, along with an energy-consistent numerical approximation. We first present the general problem in a dynamic regime, with certain types of dissipative constitutive assumptions. We then provide a numerical approximation of this problem, with the main objective of respecting energy consistency during contact in adequacy with the continuous framework. Given the presence of friction or viscosity, a dissipation of mechanical energy is expected. Moreover, we are interested in the numerical simulation of the non-smooth and non-linear problem considered, and more particularly to the optimization of Newton’s semi-smooth method and Primal Dual Active Set (PDAS) approaches. Finally, we test such numerical schemes on academic and real-life scenarios, the latter representing the contact deployment of a stainless-steel stent in an arterial tissue.
... Their interactions can be classically modeled through the Hertz theory, using non-linear damp springs, as done in the Molecular Dynamics (MD) approach (see the seminal work of Cundall [22]). In this work, we use another approach named Contact Dynamics (CD), firstly introduced by Moreau and Jean in the 1990s [23,[58][59][60][61][62]. Unlike MD, where the contact forces are modeled by functions obtained from Hertz theory, in CD, the forces are linear impulses, submitted to contact laws describing normal repulsion and tangential friction that are included into the Newton's second law of motion. ...
Granular flows occur in various contexts, including laboratory experiments, industrial processes, and natural geophysical flows. To investigate their dynamics, different kinds of physically based models have been developed. These models can be characterized by the length scale at which dynamic processes are described. Discrete models use a microscopic scale to individually model each grain, Navier-Stokes models use a mesoscopic scale to consider elementary volumes of grains, and thin-layer models use a macroscopic scale to model the dynamics of elementary columns of fluids. In each case, the derivation of the associated equations is well-known. However, few studies focus on the extent to which these modeling solutions yield mutually coherent results. In this article, we compare the simulations of a granular dam break on a horizontal or inclined planes for the discrete model convex optimization contact dynamics (COCD), the Navier-Stokes model Basilisk, and the thin-layer depth-averaged model SHALTOP. We show that, although all three models allow reproducing the temporal evolution of the free surface in the horizontal case (except for SHALTOP at the initiation), the modeled flow dynamics are significantly different, and, in particular, during the stopping phase. The stresses measured at the flow's bottom, reflecting the flow dynamics, are in relatively good agreement, but significant variations are obtained with the COCD model due to complex and fast-varying granular lattices. Similar conclusions are drawn using the same rheological parameters to model a granular dam break on an inclined plane. This comparison exercise is essential for assessing the limits and uncertainties of granular flow modeling.
... We used the non-smooth contact dynamics (CD) DEM method to simulate samples of spherical grains interacting via frictional contacts, implemented in the LMGC90 open-source platform (Dubois & Jean 2022). Compared to other DEM strategies, the CD method uses a mathematical framework in which the motion equation is implicitly integrated in a time-stepping scheme, without the need for regularisation parameters, such as contact stiffness or damping coefficients (Jean 1999). As a result, the CD method is unconditionally stable and can use larger time steps (Renouf et al. 2004). ...
... An example is the DEM (discrete element method), initially developed by Cundall et al. (1971), which considers masonry as a system of discrete blocks connected via contact points along the edges. Since then, several DEM-based calculation techniques were developed, such as the NCSM, i.e., non-smooth contact dynamics method (Jean (1999), Beatini (2017), Malomo et al. (2021)), which essentially differ from one another for the type of employed contact surfaces. Although the computational burden is reduced with the use of DEMs as compared to micro-models, it is still necessary to model a large number of elements and contact surfaces to realistically discretize a structure. ...
... The simulation of granular samples with grain size dispersion is performed using the discrete-element method known as contact dynamics (CD) (Jean, 1999;Radjai & Richefeu, 2009;Dubois et al., 2018). This method is well-suited for modelling collections of rigid bodies interacting via frictional contacts. ...
Assessing the shear strength of coarse granular soils is challenging because testing devices in the laboratory often limit the maximum particle size (d max ). While engineering standards define representative elementary volumes (REV) using the aspect ratio α=X/d max , where X is the characteristic sample size, they often disagree on the minimum α, as the effects of sample scale on shear strength are still not well understood. This paper presents a discrete-element study on the combined effect of specimen size and grading on the critical state shear strength of granular materials. The study covers a wide range of aspect ratios and demonstrates that the macroscopic response is stable for α≥15 - which is significantly higher than the standard requirement of α≥10 for simple shear tests. The granular microstructure is also strongly affected by α and the formation of column-like structures of grains carrying strong contact forces, reaching sample size independent conditions only for α≥20. Such column-like structures are shown to be primarily composed of the largest classes of grains, supporting the fact that grading has no effect on the critical state shear strength and d max correctly serves to scale a granular sample to the size of the testing device.
... Alternative approaches for addressing the dynamics of multi-block masonry systems are represented by the discrete element method (DEM) [29][30][31][32], or by time-stepping schemes investigated in [33][34][35][36][37]. The dynamic equivalence between various rocking mechanisms and a single rocking block was investigated in [38]. ...
The in-plane rocking motion of a masonry arch subjected to ground acceleration is investigated, focusing on the impacts at stereotomy sections, which may occur during the motion. It is assumed that the arch arrives at the impact moving along a prescribed four-hinge mechanism and that, after the impact, it continues its motion along a new four-hinge mechanism to be determined. The novel concept of impulse line, which is analogous to the thrust line computed during the smooth motion, is introduced to describe the impulsive stress state arising within the arch at the impact. That is the basis for extending the Housner impact model, initially proposed for the rocking motion of a free-standing column, to the more complicated case of a masonry arch behaving as a single-degree-of-freedom system. The mechanism after the impact is determined by minimizing the kinetic energy loss of the arch at impact, i.e. by maximizing its restitution coefficient, over the set of compatible mechanisms that fulfill a suitable formulation of the virtual work principle. The descending impulse line is proven to be equilibrated, kinematically admissible (i.e., not resisting the opening of the hinges after the impact), and statically admissible (i.e., corresponding to a compressive impulsive stress state). Numerical results are presented, discussing the restitution coefficient of discrete and continuous circular arches with parameterized geometry, for which the four-hinge mechanism before the impact is assumed to follow from an equivalent static analysis.
... These laws account roughly for the main features of contact and friction, and are relevant in multi-body collections where sophisticated laws cannot be captured exactly. This method was extended to deformable bodies by Jean (1999) and entitled nonsmooth contact dynamics (NSCD). See (Dubois et al. 2018) for more advanced discussions on this method. ...
Various papers have presented different methods for computations of mechanics and kinematics of discrete structural systems. However, little has been done in comparing and benchmarking the seismic response from two different discrete-system-models with experiments. This paper presents a detailed seismic performance comparison between two models—level set discrete element method (LS-DEM) and Logiciel de Mécanique Gérant le Contact (LMGC90), calibrated with shake table experiments conducted on four concrete-block configurations. Theories of both models are thoroughly explained and compared. The simulation results are benchmarked against experimental data. Finally, the principal results and significance of this benchmarking effort is discussed.
Using a discrete element method, we investigate the phenomenon of geometric cohesion in granular systems composed of star-shaped particles with 3 to 13 arms. This was done by analyzing the stability of columns built with these particles and by studying the microstructure of these columns in terms of density and connectivity. We find that systems composed of star-shaped particles can exhibit geometric cohesion (i.e., a solidlike behavior, in the absence of adhesive forces between the grains), depending on the shape of the particles and the friction between them. This phenomenon is observed up to a given critical size of the system, from which a transition to a metastable behavior takes place. We also have evidence that geometric cohesion is closely linked to the systems' connectivity and especially to the capability of forming interlocked interactions (i.e., multicontact interactions that hinder the relative rotation of the grains). Our results contribute to the understanding of the interesting and potentially useful phenomenon of geometric cohesion. In addition, our work supplements an important set of experimental observations and sheds light on the complex behavior of real, three-dimensional, granular systems.
We introduce an approach to particle breakage, wherein the particle is modeled as an aggregate of polyhedral cells with their common surfaces governed by the Griffith criterion of fracture. This model is implemented within a discrete element code to simulate and analyze the breakage behavior of a single particle impacting a rigid plane. We find that fracture dynamics involves three distinct regimes as a function of the normalized impact energy ω. At low values of ω, the particle undergoes elastic rebound and no cracks occur inside the particle. In the intermediate range, the particle is damaged by nucleation and propagation of cracks, and the effective restitution coefficient declines without breakup of the particle. Finally, for values of ω beyond a well-defined threshold, the particle breaks into fragments and the restitution coefficient increases with ω due to kinetic energy carried away by the fragments. We show that particle damage, restitution coefficient, and fracture efficiency (the amount of energy input consumed for particle fracture) collapse well as a function of dimensionless scaling parameters. Our data are also sufficiently accurate to scale fragment size and shape distributions. It is found that fragment masses (volumes) follow a power-law distribution with an exponent decreasing with fracture energy. Interestingly, the average elongation and flatness of fragments are very close to those observed in experiments and lunar samples at the optimal fracture efficiency.
In many geological hazards, such as landslides, a large number of irregular blocks start moving. Their interaction on the way down renders prediction of disaster scopes difficult. To study this process and to provide a novel method for validation and calibration of numerical tools for its simulation, a cube movement test is designed. The goal of this research is to obtain patterns of movement of cubes, starting from different initial stacking arrangements. Cubes of four sizes are inserted into a hollow cylinder. Their distribution after lifting the cylinder is determined. Three categories of tests refer to three different strategies of filling the cubes into the cylinder. In order to simulate cube movement tests, a numerical tool is developed in the framework of the continuum–discontinuum element method (CDEM). The contact between the individual cubes is modeled by the contact‐pairs‐based algorithm. Both the contact state and type are detected by determining the half‐space relation between contact pairs. The final positions of the cubes are strongly related to their initial arrangement. The latter is different in every test, even if the same strategy is used to fill the cubes into the cylinder. It is found that at least 20 experiments/simulations are required to obtain statistically representative results. The new test provides valuable data for validation of numerical tools used for the simulation of mass movement processes. The proposed numerical method captures the complicated movements of blocks.
In-space assembly (ISA) has become a potential solution to building large space structures. However, the motion of an ISA system will experience a sudden topology change when two flexible modules are assembled together thus causing the whole system to oscillate. Therefore, understanding the dynamics of an ISA system during and after assembly is of importance. This paper presents a methodology for simulations of ISA system based on the modeling of flexible multibody system with topology changes. The theory of Absolute Nodal Coordinate Formulation (ANCF) is applied to describing the deformation of flexible bodies. Meanwhile, combining the impulse-momentum equations and the constraint equations in velocity level, a linear algebraic equation is obtained to solve the discontinuous velocity changes when different modules are assembled together. In addition, the dynamic model of a typical ISA system made up of hexagon truss modules is established based on Lagrange’s equations of the first kind. Finally, numerical simulations are provided to evaluate the effect of gravity and to investigate a basic assembly procedure. The presented model is verified to be able to accurately describe the assembly procedure and able to capture the oscillations after topology changes. The simulation results are furthermore analyzed to study the dynamic characteristics of the ISA system.
This contribution introduces a methodology for the detection of isolated branches of periodic solutions to the nonlinear mechanical equation of motion for systems featuring frictionless contact interfaces. This methodology relies on a harmonic balance method-based solving procedure combined with the application of the Melnikov energy principle. It is able to predict the location of isolated branches of solutions in the vicinity of families of autonomous periodic solutions, namely nonlinear normal modes. The methodology is first applied on a two-degree-of-freedom phenomenological system in order to illustrate its relevance and accuracy. In particular, for this academic application, the proposed methodology yields an understanding of the discontinuous evolution of the first nonlinear resonance frequency as the forcing amplitude increases. Isolated branches of solutions featuring high amplitudes of vibration are detected far beyond nonlinear resonance frequencies obtained with a typical application of the harmonic balance method coupled with an arc-length continuation algorithm. Demonstration of the applicability of the proposed methodology to high-dimensional industrial finite element models is made with the nonlinear vibration analysis of a transsonic compressor blade, NASA rotor 37, subjected to an harmonic loading and blade-tip/casing structural contacts. The proposed methodology yields numerous isolated branches of solutions, which relevance is assessed by means of time integration simulations. In the end, the presented results underline that typical continuation algorithms may yield a significant underestimation of nonlinear resonance frequencies.
By means of two-dimensional numerical simulations based on contact dynamics, we present a systematic analysis of the joint effects of grain shape (i.e., grain elongation) and system size on silo discharge for increasing orifice sizes D. Grains are rounded-cap rectangles whose aspect ratio are varied from 1 (disks) to 7. In order to clearly isolate the effect of grain shape, the mass of the grains is keeping constant as well as the condition of the discharge by reintroducing the exiting grains at the top of the silo. In order to quantify the possible size effects, the thickness W of the silos is varied from 7 to 70 grains diameter, while keeping the silos aspect ratio always equal to 2. We find that, as long as size effects are negligible, the flow rate Q increases as a Beverloo-like function with D, also for the most elongated grains. In contrast, the effects of grain elongation on the flow rate depend on orifice size. For small normalized orifice sizes, the flow rate is nearly independent with grain elongation. For intermediate normalized orifice sizes the flow rate first increases with grain elongation up to a maximum value that depends on the normalized size of the orifice and saturates as the grains become more elongated. For larger normalized orifice size, the flow rate is an increasing function of grains' aspect ratio. Velocity profiles and packing fraction profiles close to the orifice turn out to be self-similar for all grain shapes and for the whole range of orifice and system sizes studied. Following the methodology introduced by Janda et al. [Phys. Rev. Lett. 108, 248001 (2012)], we explain the nonlinear variation of Q with grain elongation, and for all orifice sizes, from compensation mechanisms between the velocity and packing fraction measured at the center of the orifice. Finally, an equation to predict the evolution of Q as a function of the aspect ratio of the grains is deduced.
This article presents the development and calibration of a numerical model simulating the response of a novel rockfall protection structure subjected to localized dynamic loading. This structure is made of piled-up concrete blocks interconnected via metallic components whose dynamics response under projectile impact is examined via real-scale experiments. The corresponding numerical model is developed in a python based open source software Siconos which implements the Non-Smooth Contact Dynamics (NSCD) method. The geometrical features and mechanical properties are incorporated in the model via specific developments pertinent to the modelling requirements in Siconos. Some parameters peculiar to the numerical model are calibrated against the spatial–temporal measurements from two full-scale impact experiments. The Bayesian interface statistical learning method aided by the polynomial chaos expansion based meta-model of the NSCD model is deployed for the calibration. The additional understanding of the model dynamics through the byproducts of the meta-model is highlighted. In the end, the NSCD model is successfully calibrated against the spatial–temporal response of the experimental structure with more than 90% accuracy for impact energies up to 1 MJ.
Internal parameters of non-smooth systems, such as hysteresis and gap systems, are unable to be measured when the system is enclosed. For a non-smooth gap system, the actual system state is difficult to be evaluated due to state transitions and the coupled internal mechanical and geometric parameters. Moreover, the system state evaluation directly affects the parameter estimation result, which, in turn, will affect the evaluation of the system state at the next time instant. The interdependence between the estimated parameters and the subsequent state evaluation creates a large challenge for the traditional filter method. Based on the idea of estimating system parameters at different states from the discontinuous extended Kalman filter (DEKF) and the discontinuous unscented Kalman filter (DUKF), a parameter estimation algorithm, namely DBAA-DUKF, is proposed herein by introducing a dynamic boundary approximation algorithm (DBAA) to address this contradiction. Firstly, three evaluation conditions are presented for the system state identification. Then, DBAA-DUKF adjusts the state transition parameters and their upper and lower limits according to the difference between the evaluation results of the system state and filtering results. The parameter estimation accuracy of DBAA can be improved by adjusting the upper and lower limits of state transition parameters to constrain them in a narrow range. Finally, a single-sided nonlinear impact system with a significant gap and a double-sided gap system with small gaps are used to verify the effectiveness and feasibility of the algorithm. The results show that DBAA-DUKF can effectively solve the parameter estimation problem in the uncertain state transition conditions of non-smooth gap systems and can obtain better estimation results than DEKF and DUKF.
This work considers a contact problem with friction involving one contact point and two degrees-of-freedom. The contacting structure is linear elastic. Two different models of contact interaction are considered, the classical Signorini unilateral contact law and a normal compliance law. Coulomb's law of friction is used. All possible so-called rate problems are solved, from which one concludes that the quasistatic problem may possess non-uniqueness and non-existence of solutions. In the case of the normal compliance law this can be explained by a softening structural response. For Signorini's law softening explains only some of the possible situations where non-uniqueness can occur.In dieser Arbeit wird ein Kontaktproblem mit Reibung behandelt, das einen Kontaktpunkt und zwei Freiheitsgrade einschliet. Die kontaktgebende Struktur ist linearelastisch. Zwei verschiedene Modelle der Kontaktwirkung sind bercksichtigt: Erstens das klassische einseitige Signorini-Kontaktgesetz und zweitens ein Gesetz fr die Nachgiebigkeit in Normalenrichtung. Das Coulombsche Reibungsgesetz wird verwendet. Alle mglichen sogenannten Geschwindigkeitsprobleme werden gelst, woraus geschlossen wird, da das quasistatische Problem Nichteindeutigkeit und Nichtexistenz der Lsung besitzen kann. Im Fall des Nachgiebigkeitsgesetzes kann dieses als abfallende Struktursteifigkeit erklrt werden. Im Fall eines Signorini-Gesetzes erklrt dieses nur einige der mglichen Situationen, wo Nichteindeutigkeit auftreten kann.
Consider a point moving with respect to a rigid body. The system made up of these two elements is deformable since the distance of the point and the body changes. Because the system is deformable, we define strain rates and interior forces. The latter are percussions and forces defined by their virtual work. The equations of motion are derived from the principle of virtual work. The constitutive laws for the interior percussions and forces are the other equations which describe the evolution of the system, in particular the collisions which occur between the point and the body. Examples illustrate the ability of the theory to analyse practical situations.
Contact and friction phenomena are common and very important in many fields of mechanical engineering. Recently, a significant research effort has been made to develop effective computational methods for finite element analysis of nonlinear problems in structural dynamics. This work may contribute to the discussion about different aspects of the implicit contact dynamics analysis. We begin with a brief presentation of the standard Newmark and Newton-Raphson approach. We propose several modifications to these schemes, presenting a set of first-order integration algorithms, and analyzing their numerical characteristics and utility in the numerical treatment of contact.
This chapter presents general numerical methods for treating dynamical problems involving unilateral contact and dry friction. Some examples of applications related to the structural response of rigid or deformable geomaterials, such as rocks, soils, collections of blocks, granular materials, are illustrated. The chapter explains the Coulomb's dry friction law. This law is relevant for a large class of applications to geomaterials. It accounts for the main features of dry friction. It may be easily improved without drastic changes in the proposed methods. The frictional problems appear to be strongly non linear, and call for the techniques of nonsmooth mechanics. Convex analysis is widely used to formulate friction equations and numerical algorithms.
One considers a continuous medium coming into contact with a rigid obstacle or another deformable body. Quasi-static evolution problems (i.e. with negligible inertia) are considered as well as proper dynamical problems. Formulations of unilateral contact are proposed for these two cases. Dry friction is taken into account through Coulomb’s law. A system of equations for the time and space discretized problem is proposed together with an algorithm for solving this system. The derivation of equations when performing space variable discretization is specially developed in this paper.
Three different so-called rate boundary-value problems of frictional contact between a linear elastic body and a rough stiff foundation are derived in this paper. The investigation starts from the principle of virtual power, balance of energy and the second law of thermodynamics. Conditions for the existence of unique solutions for the rate problems are derived.
The dynamical problem for a viscoelastic body involving unilateral contact and Coulomb friction is formulated which takes into account discontinuities of relative velocities when impacts occur. We propose several first-order implicit numerical schemes to solve the time discretized equations. The results are compared to those obtained by a higher-order standard numerical schemes. The discrete problem, with the unknown velocity, is reformulated as a complementarity problem and is then solved by Lemke’s mathematical programming method.
The use of the classical Coulomb law of friction in the formulation of contact problems in elasticity leads to both physical and mathematical difficulties; the former arises from the fact that this law provides a poor model of frictional stresses at points on metallic surfaces in contact, and the latter is due to the fact that the existence of solutions of the governing equations can be proved only for very special situations. In the present report, nonclassical friction laws are proposed in an attempt to overcome both of these difficulties. A class of contact problems is considered involving the equilibrium of linearly elastic bodies in contact on surfaces on which nonlocal and nonlinear friction laws are assumed to hold. The physics of friction between metallic bodies in contact is discussed and arguments in support of the theory are presented. Variational principles for boundary-value problems in elasticity in which such nonlinear nonlocal laws hold are then developed. A brief discussion of the questions of existence and uniqueness of solutions to the nonlocal and nonlinear problems is given. (from authors' abstract)
An approach to the dynamics of mechanical systems with a finite number of degrees of freedom, involving unilateral constraints, is developed. In the n-dimensional linear spaces of forces and velocities, some classical concepts of Convex Analysis are used, but no convexity assumption is made concerning the constraint inequalities. The velocity is not supposed to be a differentiable function of time, but only to have locally bounded variation, so the role of the acceleration is held by a n-dimensional measure on the considered time interval. Dynamics is then governed by measure differential inclusions, which treat possible velocity jumps on the same footing as smooth motions. Possible collisions are described as soft, thus dissipative. Friction is taken into account under a recently proposed expression of Coulomb’s law. These formulations have the advantage of generating numerical algorithms of time-discretization, able to handle, in particular, the nonsmooth effects arising from unilaterality and from dry friction.
THEORY. Multibody Kinematics. Dynamics of Rigid Body Systems. Contact Kinematics. Multiple Contact Configurations. Detachment and Stick-Slip Transitions. Frictionless Impacts by Newton's Law. Impacts with Friction by Poisson's Law. The Corner Law of Contact Dynamics. APPLICATIONS. Applications with Discontinuous Force Laws. Applications with Classical Impact Theory. Applications with Coulomb's Friction Law. Applications with Impacts and Friction. References. Index.
The mechanical system we propose to deal with is a finite set of perfectly rigid bodies submitted to: (a) usual constraints, meaning holonomous, bilateral, frictionless constraints, depending or not on time; (b) forces, they are functions of time, velocity, and the generalized variables of the system; (c) punctual contacts with dry friction between some bodies of the system or between bodies of the system and some extraneous rigid bodies, the motion of which is explicitly given as a function of time. In the first part, the general equations of the problem are introduced, existence theorems are given, and comments are made about the mechanical significance of both the assumptions and the foregoing results. The equations are set in the second part, which is thus essentially concerned with mechanics and may be read independently. The proofs of the theorems are to be found in the third part.
We present a finite element method for a class of contact-impact problems. Theoretical background and numerical implementation features are discussed. In particular, we consider the basic ideas of contact-impact, the assumptions which define the class of problems we deal with, spatial and temporal discretizations of the bodies involved, special problems concerning the contact of bodies of different dimensions, discrete impact and release conditions, and solution of the nonlinear algebraic problem. Several sample problems are presented which demonstrate the accuracy and versatility of the algorithm.
The distinct element method is a numerical model capable of describing the mechanical behavior of assemblies of discs and spheres. The method is based on the use of an explicit numerical scheme in which the interaction of the particles monitored contact by contact and the motion of the particles modelled particle by particle. The main features of the distinct element method are described. The method is validated by comparing force vector plots obtained from the computer program BALL with the corresponding plots obtained from a photoelastic analysis.
A C0 three-node shell finite element well suited to non-linear calculations is proposed. The element is based on Mindlin kinematics and the degenerated solid approach. Linear Lagrange functions are used for geometry and displacement interpolations. The formulation is made in the natural material frame. A strain interpolation avoids shear locking and an intermediate material frame related to the element sides is introduced in order to fix nodal transverse shear strain components. The modifications of strain interpolations concern both the non-linear and linear parts of strain and are taken into account in ail calculations, among others in the expression of the initial stress stiffness matrix. A single set of integration points on the normal at the centre of gravity is sufficient, which is very interesting for numerical efficiency especially in the case of non-linear analyses.
In this paper a new time-stepping method for simulating systems of rigid bodies is given which incorporates Coulomb friction and inelastic impacts and shocks. Unlike other methods which take an instantaneous point of view, this method does not need to identify explicitly impulsive forces. Instead, the treatment is similar to that of J. J. Moreau and Monteiro-Marques, except that the numerical formulation used here ensures that there is no inter-penetration of rigid bodies, unlike their velocity-based formulation. Numerical results are given for the method presented here for a spinning rod impacting a table in two dimensions, and a system of four balls colliding on a table in a fully three-dimensional way. These numerical results also show the practicality of the method, and convergence of the method as the step size becomes small.
In this paper, a formulation is presented for the finite element treatment of multibody, large deformation frictional contact problems. The term multibody is used to mean that when two bodies mechanically contact, both may be deformable. A novel aspect of the approach advocated is that the equations governing contact are developed in the continuum setting first, before deriving the corresponding finite element equations This feature distinguishes the current work from many earlier treatments of contact problems and renders it considerably more general. In particular, the approach yields a characterization of the frictional constraint (assuming a Coulomb law) suitable for arbitrary discretizations in either two or three dimensions. A geometric framework is constructed within which both frictionless and frictional response are naturally described, making subsequent finite element discretization a straightforward substitution of finite-dimensional solution spaces for their continuum counterparts. To our knowledge, this general formulation and implementation of the frictional contact problem in a finite element setting has not been reported previously in the literature.
The development includes exact linearization of the statement of virtual work, which enables optimal convergence properties for Newton-Raphson solution strategies, and which appears to be highly desirable (if not essential) for the general robustness of implicit finite element techniques. Since the theory and subsequent linearization require no limitations on the amount of deformation or relative sliding that can occur, the resulting treatment of frictional contact is suitable for a wide range of examples displaying significant non-linear behaviour. This assertion is substantiated through presentation of a variety of examples in both two and three dimensions.
The sweeping process, introduced some time ago by the author with motivation in plasticity theory, today remains an object of mathematical research. It is considered in this paper as the prototype of an evolution conditioned by inequality constraints. Since the governing differential requirements are only of order one with respect to time, this provides a simplified setting for analysing some numerical and theoretical features also present in unilateral dynamics. The latter is governed by differential inclusions of order two, for the numerical handling of which the existing literature proposes diverse strategies, briefly discussed. The paper is especially intended to offer an introduction to the numerical approach called 'contact dynamics'.
This paper addresses the general problem of formulating continuum models of a large class of dynamic frictional phenomena and of developing computation methods for analyzing these phenomena. Of particular interest are theories which can adequately predict stick-slip motion, frictional damping in structural dynamics, and sliding resistance. This work is divided into three principal parts. In Part I, a large body of experimental and theoretical literature on friction is critically reviewed and interpreted as a basis for models of dynamic friction phenomena. In Part II, continuum models of interfaces are developed which simulate key interface properties identified in Part I. Variational principles for a class of dynamic friction problems are also established. In Part III, finite element models and numerical algorithms for analyzing dynamic friction are presented. Also, a dynamic stability analysis is presented in which it is established that stick-slip motion can be associated with dynamic instability of the governing nonlinear system for certain ranges of slip velocity and coefficient of friction. Numerical results suggest that the new models derived here can satisfactorily depict a large and important class of dynamic friction effects.
The numerical treatment of contact problems with friction is considered. Starting from the algebraic stiffness equation of two discretized linear elastic bodies and taking the contact and friction laws into account we derive a new type of Linear Complementarity Problem (LCP). A principal pivot algorithm for this problem is suggested. It is then shown how the evolution of displacements and contact stresses for a prescribed load history is obtained. As an application of the method we solve a three-dimensional punch indentation problem.
The existence of solutions of rigid body dynamics with impact, shock and Coulomb friction in the sense of measure differential inclusions is announced. For the first time, such results include the famous counter-examples of Painlevé. The resolution of the paradox involves impulsive forces without collisions, and the use of the post-impact time in the formulation of Coulomb's law.RésuméL'existence de solutions au problème de la dynamique des corps rigides incluant impacts, chocs et frottement coulombien au sens des inclusions de mesures différentielles, est annoncée. Ces résultats englobent pour la première fois les célèbres contre-exemples de Painlevé. La résolution du paradoxe fait usage de forces impulsives sans collision, et du temps de post-impact dans la formulation de la loi de Coulomb.
A mixed penalty-duality formulation of the frictional contact problem, inspired from an augmented Lagrangian approach is proposed. The continuity of the resulting conewise linear operator is used to establish a uniqueness condition on the coefficient of friction. Modified and generalized Newton methods are examined and sufficient conditions for their convergence conjectured. A cylindrical frictional contact problem assesses the stability of the method. Mixed penalty-duality methods are found more accurate and stabler than penalty methods and as economical as them.
The purpose of this work is to present a family of implicit numerical methods devoted to frictional contact problems in dynamic deep drawing simulations. These methods have been developed to overcome the convergence problems due to strong non-linearities and to simulate spring-back effects. They are based on quasi-Newton solver types. To ensure the convergence of these quasi-Newton methods, time step conditions are required. Then, several convergence criteria are developed to estimate the critical time step and are compared with the stability criteria of the explicit schemes. A relationship between these convergence criteria and the stability one is discussed. The algorithmic performances of these implicit methods are numerically estimated and computational results on a benchmark test sensible to spring-back effects are presented.
The purpose of this work is to present some mathematical and numerical results concerning an implicit method devoted to frictional contact problems. It has been implemented in a dynamical deep drawing simulation software (SIMEM3 Renault) where unilateral contact and dry friction is assumed between the metal sheet and the tools. The method may be viewed as a nonlinear block Gauss-Seidel algorithm. A convergence theorem is proved using nonsmooth analysis theory. Several numerical results are presented to illustrate the behaviour of this algorithm.
This article reviews the numerical methods used for a few years in the program TACT to solve contact problems with non-associated Coulomb’s friction. These methods include: a penalty method to enforce the contact and adherence conditions respectively, an implicit projection method to integrate the slip rule, the finite element method for the spatial discretization and a generalized Newton method to overcome the contact and friction nonlinearities. Recent advances improving the robustness of the resulting frictional contact algorithm are reported. In particular, a necessary and sufficient condition on the friction coefficient for the solution to flat contacts to be unique is stated and a damping factor is introduced to guarantee the algorithm convergence to this solution in the two-dimensional case. The flat punch problem is used to illustrate both the accuracy and efficiency of the method.
Relying on contact dynamics simulations, we study the statistical distribution of contact forces inside a confined packing of circular rigid disks with solid friction. We find the following: (1) The number of normal and tangential forces lower than their respective mean value decays as a power law. (2) The number of normal and tangential forces higher than their respective mean value decays exponentially. (3) The ratio of friction to normal force is uniformly distributed and is uncorrelated with normal force. (4) When normalized with respect to their mean values, these distributions are independent of sample size and particle size distribution.
Programmation mathématique pour le contact avec frottement et comparaison avec d'autres méthodes
- Chabrand
P. Chabrand, F. Dubois and M. Raous, Programmation math~matique pour le contact avec frottement et comparaison avec d'autres m&hodes, in: Actes du 2bme Colloque National en Calcul des Structures (Hermes, 1995).
Numerical dynamics for the simulation of deep drawing
- M Jean
- F Jourdan
- B Tathi
M. Jean, F. Jourdan and B. Tathi, Numerical dynamics for the simulation of deep drawing, in: Proc. of IDDRG 1994, Handbooks on Theory and Engineering Applications of Computational Methods, May 16-17, 1994.
Unilaterality and dry friction in the dynamics of rigid bodies collections
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Numerical dynamics for the simulation of deep drawing
- Jean