The Finite Element Method
Abstract
The sixth editions of these seminal books deliver the most up to date and comprehensive reference yet on the finite element method for all engineers and mathematicians. Renowned for their scope, range and authority, the new editions have been significantly developed in terms of both contents and scope. Each book is now complete in its own right and provides self-contained reference; used together they provide a formidable resource covering the theory and the application of the universally used FEM. Written by the leading professors in their fields, the three books cover the basis of the method, its application to solid mechanics and to fluid dynamics.* This is THE classic finite element method set, by two the subject's leading authors * FEM is a constantly developing subject, and any professional or student of engineering involved in understanding the computational modelling of physical systems will inevitably use the techniques in these books * Fully up-to-date; ideal for teaching and reference
... More specifically, for a 2D four-node element using mapped bilinear shape functions, a necessary condition is that any internal angle should not be greater than, theoretically, 180 • . In numerical implementation, the determinant of the Jacobian matrix should be always checked for its positivity to avoid severely distorted elements [1,13,24]. In addition, in the numerical integration of Galerkin weak form, Gauss quadrature is a commonly used method. ...
... In Scheme 3 (S3), SC/GP=1 is used all the time and energy norm. The displacement norm is given in Eq. (24) and the energy norm is defined by In the computations, the nodes on the left boundary are constrained using the exact displacements obtained from Eq. (26) and the loading on the right boundary uses the distributed parabolic shear stresses in Eq. (27). The beam is analyzed using different number of elements and smoothing cells. ...
... Then we increase the number of smoothing cells or Gauss points in FEM and study the relative error of temperature using Eq. (24). From the results listed in Table 8, we notice that, when GP/SC = 4, the temperature obtained by SFEM is much more accurate than that of FEM. ...
... Burbridge and Awruch [3] applied this scheme using hexahedral finite elements with analytical evaluation of the element matrices but evaluating the inverse matrix and the determinant of the Jacobian matrix at the center of the element, together with a shock capturing technique for the solution of viscous and non-viscous compressible laminar flows [4]. However, for subsonic regimes (with M <0.9) or for incompressible flows, the TG scheme presents certain instabilities, as reported by Zienkiewicz and Taylor [5]. ...
... Step CBS algorithms are mentioned but the respective equations are not shown for the sake of brevity. The reader can look at the references [1,3,5,6,8] to get more details. Instead, the full matrix equations for the proposed MTG scheme are presented in the section 4. ...
... A conditionally stable explicit time integration version of the TG scheme [4,5,15], may be obtained working with a lumped mass matrix (where only elements of the main diagonal are different from zero). ...
As already demonstrated by different authors, the Taylor-Galerkin (TG) scheme, in the context of the Finite Element Method (FEM), is particularly suitable for the solution of supersonic flows. The TG scheme, using hexahedral finite elements with analytical evaluation of element matrices, is applied in this work. Tools to avoid locking and a shock capturing technique for the solution of supersonic viscous and non-viscous compressible flows are also employed. However, TG scheme usually presents instabilities in subsonic flows. Even in cases in which the free stream Mach number corresponds to supersonic flows, there are always flow regions, specifically near the walls of the immersed obstacles, where the speed is lower and the local Mach number corresponds to a subsonic flow. The Characteristic Based Split (CBS) scheme was developed in order to obtain a single method to improve the behavior with respect to TG method in subsonic and supersonic regimes. In the last two decades some works have shown advantages in convergence rates of the CBS method when compared to the TG algorithm. However, simulation time increases in the CBS method since split operations, typical of this algorithm, imply in additional element loops. In this paper a hybrid algorithm called Modified-Taylor-Galerkin scheme (MTG) is proposed. This algorithm presents advantages with respect to TG and CBS schemes in terms of convergence properties and computational processing time. In order to get an efficient algorithm, the element matrices are analytically integrated. This is performed with two different approaches. In the first approach the inverse matrix and the determinant of the Jacobian matrix at element level are evaluated with a reduced integration form, using the point located in the center of the element for mass, convective, diffusive and stabilization element matrices; all these matrices are integrated analytically. In the second approach, mass and convective matrices are calculated by a complete integration scheme (including the inverse matrix and the determinant of the Jacobian matrix at element level in the analytical expression to be integrated) and the diffusive and stabilization matrices are calculated with a reduced integration form, using the point located at the center of the element to calculate the inverse matrix and the determinant of the Jacobian matrix at element level. Finally, this work incorporates the Spalart-Allmaras (S-A) turbulence model using a conservative version of the transport equation, as proposed by the authors of the original S-A model in a later paper. Algorithms are tested to determine convergence rate improvements in both laminar and turbulent cases and for different Mach numbers (supersonic, transonic and subsonic flows).
Keywords:
CompressibleTurbulentFEMTaylor-GalerkinCBSSpalart-Allmaras
... To study the finite element method is a challenging task and different approaches are available. This ranges from classical lectures and the corresponding textbooks [2,3,15,22,23] to the classical tutorials with 'hand calculations' [9]. In a more modern academic context, so-called problem or project based approaches are also common in some countries [16]. ...
... The general expression for the elemental stiffness matrix K e of a two-noded rod 1 element with cross-sectional area A(x), Young's modulus E(x), and length L can be expressed as [2,22] ...
... The general expression for the elemental stiffness matrix K e of a two-noded EulerBernoulli beam 1 element with second moment of area 2 I (x), Young's modulus E(x), and length L can be expressed as [2,22] ...
This chapter introduces first the theory to derive the elemental stiffness matrix of Euler–Bernoulli beam elements. Then, the principal finite element equation of such beams and their arrangements as plane frame structures are briefly covered. Furthermore, a combination of Euler–Bernoulli beam and rod element is introduced as a generalized beam and frame element. Comments on the computer implementation of the corresponding Maxima modules are provided. The chapter includes detailed Maxima examples which allow an easy transfer to other problems.
... The mixed formulation has been proposed to do exactly this, that is, to model the acoustic-mechanical interaction problems without having an explicit boundary representation. The mixed finite element formulation, also called the u/p (displacement/pressure) formulation, can be found in many references (see, e.g., Zienkiewicz and Taylor (2000)). In Wang and Bathe (1997), the formulation (2007), the static mixed FE-formulation was used to solve pressure load problems in density-based topology optimization and in Yoon et al. (2007), the formulation was used for the first time for topology optimization of acoustic-structure interaction problems. ...
... In order to realize a stable finite element solution to the mixed u/p formulation, displacement variables should use higher order interpolations than the auxiliary pressure variable (Zienkiewicz and Taylor 2000;Wang and Bathe 1997). To this end, displacement variables are discretized with second order Lagrangian shape functions whereas the pressure field is represented using first order Lagrangian shape function. ...
... The density-based design is thresholded at γ = 0.5 using the marching square algorithm and the resulting frequency responses are collected in Fig. 12. From the plot, it is clear that the performance of the density-based design is practically identical to the best result obtained using the level set method. Hence, the discrepancy in objective value is due to the mixed formulation requiring a finer mesh than the segregated analysis (Zienkiewicz and Taylor 2000). Figure 13 shows the sound pressure level (SPL) of the acoustic domain. ...
The pursuit for design improvements by geometry modifications can easily become prohibitive using a trial and error process. This holds especially when dealing with multi-physics problems—such as acoustic-structure interaction—where it is difficult to realize design improvements intuitively due to the complexity of the coupled physics. Compared to classical shape optimization, where a near optimal shape has to be supplied as an initial guess, topology optimization allows for innovative designs through a completely free material distribution, such that the topology can change during the optimization process. The goal of this article is to provide a comprehensive critical review of the proposed strategies for topology optimization of coupled acoustic-structure interaction problems. The work includes a comparison of topology optimization formulations with density, level set, and evolutionary-based methods and discusses the corresponding strengths and weaknesses through the considered application examples. The review concludes with recommendations for future research directions.
... The concept of spatial discretization is at the core of the finite element method (FEM), see Bathe [10], Belytschko et al. [18], Hughes [74], Strang et al. [135], and Zienkiewicz et al. [152]: the functional spaces for the description of the different fieldse.g. geometry, displacements, stressesare approximated by finite elements with their locally confined basis functions. ...
... geometry, displacements, stressesare approximated by finite elements with their locally confined basis functions. Following the isoparametric paradigm (see Strang et al. [135] and Zienkiewicz et al. [152]) the solution field is expressed through the same ansatz as the geometry itself. ...
... The quality of the approximation can generally be improved by refinement, i.e. either increasing the number of elements (h -method) or using highervalued shape functions (p -method), see Hughes [74] and Zienkiewicz et al. [152]. ...
Methods for the design, analysis and verification of lightweight structures considering their non-linear behavior are developed. To perform finite element analyses (FEA) on CAD models, the isogeometric B-Rep analysis (IBRA) is extended to the form-finding and analysis of structural membranes. Configuration update methods are developed to consider the mounting process. Effects of the non-linear behavior on the verification of safety are investigated and contributions to a future Eurocode for structural membranes are made.
... The body is infinitely long in the 3 Ox direction. Since the lineal force extends to infinity in this direction, however, the plane deformation state arises in the 12 ...
... It is seen from Eq. obtained, proving the validity of the functional in (11). Now, the FEM model for the considered problem is constructed using the virtual work principle and the Ritz method [12]. To do this, the domain D is split into a finite number of smaller piecewise consisting of ninenode smooth rectangular elements. ...
In this study, the frequency response of a pre-stressed slab, which stands on a rigid foundation, subject to a timely harmonic loading was considered. The investigation is implemented according to the piecewise homogeneous body model utilizing the three-dimensional linearized theory of elastic waves in initially stressed bodies (TLTEWISB). The considered body was designed joining to three discrete slabs side-by-side. It was assumed that there exists a rigidly clamping state at all interface planes on the system. A mathematical model of the problem is constructed and the system related to equations of motion is numerically solved using the finite element method (FEM). Particularly, the effect the ratio of the layer length has on the frequency response of the slab was presented.
... With the help of the above coefficients, the finite element equations can be written in the following type: Using Newton-Raphson method explained by Reddy [48], the obtained nonlinear Eqs. (51) to (55) are converted into linear algebraic equations. Finally, these linear equations are solved by employing triangular factorization method and reduced integration method expressed by Zeinkiewicz et al. [49]. ...
... The convergence criterion of the numerical solution along with error estimation has been set to | | m+1 − m| | ≤ 10 −5 , where m is the number of iteration and is a function of U, V and θ. The application of this simulation is well described by Taylor and Hood [50], Zienkiewicz and Taylor [51] and Dechaumphai. [52]. ...
In this paper, a computational study of natural convection in a grooved enclosure filled with water-based nanofluid in the presence of external magnetic field is numerically investigated. Two-component non-homogeneous model is introduced to develop the governing partial differential equations. Galerkin finite element method is used to solve the governing equations. The computation is carried out for a wide range of governing parameters such as Rayleigh number (103 ≤ Ra ≤ 106), magnetic field parameter (10 ≤ Ha ≤ 100) and volume fraction of nanoparticles (0% ≤ ϕ ≤ 5%) with fixed values of remaining parameters. A detailed parametric analysis is performed to show the effects of physical parameters on the fluid flow and temperature distributions within the enclosure via streamlines, isotherms, isoconcentrations, mid-sectional velocities, average Nusselt number and temperature, respectively. In addition, the entropy generation and Bejan number are also computed and discussed elaborately. The results of the current study are compared to those of previous numerical and experimental studies and found to be in rational agreements. The results ascertain that the average Nusselt number and entropy generation increase with rising Rayleigh number and nanoparticle volume fraction, whereas they decrease with increasing magnetic field strength. Moreover, it is found that the appropriate combination of governing parameters can maximize the heat transfer rate and minimize the entropy generation as well.
... R= Ct=N[ =D OvDN=OQB Ovv=t l}DUq sm = QD p@=kQ}e O=wt |UQQ@ x@ [10] "CU= xOW \ki =yu; C=ar=]t "Ovvm u}}aD = Q K]Ut VvQm Cr=L QO xDi=} pmWQ}}eD |=yu=tr= |=yVvQm =@ p=Ut \@=wQ '|rm Qw] x@ "CU= xOW OwOLt sm = QD p@=kQ}e O=wt x@ u}=Q@=v@ "OvDU}v TQDUO QO ?re= w CU= xO}J}B Q=}U@ |wDUt Cr=L QO nQR@ |=yu=tr= |UQQ@ x@ Q[=L Q=DWwv u=oOvU}wv |xkqa 'xOWxQ=W= j@=wU x@ xHwD =@ sQJ 'l}DUq Ovv=t u=UWmQ@= O=wt Q=DiQ |UQQ@ w nQR@ |=ypmWQ}}eD =@ |wDUt "CU= xOW ?rH """ w u=UWmQ@= Q=DiQ =@ |]NQ}e |wDUt |=yu=tr= |Ov@pwtQi 'Q[=L Q=DWwv QO p}rLD |= Q@ pt=m Sv=Qo q |Ov@pwtQi R= w x=Q= nQR@ |=yVvQm =@ |=yxR=U |= Q@ x@ xt=vQ@ "CU= xOW KQ]t u=UWmQ@= O=wt \@=wQ u}vJty "CU= xOW xO=iDU= p=Ft wO w CU= xOW xDWwv u=UWmQ@= p=Ut |]NQ}e p}rLD |= Q@ 7 ?rDt u=@R u}vJty "CU= xOW xU}=kt C=ar=]t Qo}O G}=Dv =@ w pL |O=yvW}B |Ov@pwtQi =@ xDWwv fqg e w fg e`w tHt CQwYx@ Ov=wD|t u=tr= |}=Hx@=H u=O}t u}=Q@=v@ %11 |x]@= Q OwW (11) w u 'fqg e =@ \@DQt |QwQ[ |U=U= |}=Hx@=H u=O}t 0 w u 0 QO xm w pmW`@=D T} QD=t [N q ] 'fg e =@ \@DQt |xOWxi=[= |rN=O |}=Hx@=H u=O}t %OvDUy 12 |x]@= Q pmW`@=D T} QD=t |rN=O |=yQDt= Q=B [N ] ...
... wO pt=W R}v w 2 '1 |rN=O QDt=Q=B wO pt=W u |}=Hx@=H u=O}t 'xwqax@ %OvDUy 10 |x]@= Q CQwYx@ fg e Q=OQ@ |=yQDt= Q=B "CU= 0 2 w 0 1 |rN=O QDt=Q=B fg e = [1 0 1 2 0 2 ] T (10) P = 3(L 3 L 1 )(L 4 L 2 ) (g 2 g 3 )(L 3 L 1 ) (g 1 g 2 )(L 4 L 1 ) 1 2 (g 2 g 4 g 1 g 3 ) 1+g 1 g 3 +g 2 g 4 (14) [19] '1993 p=U QO |WywSB %CU= xOQm hPL 29 |x]@= Q CQwYx@ swO E ij = 1 2 (u 0 i;j + u 0 j;i + u 0 m;i u 0 m;j ) + 1 2 (u i;j + u j;i ) = 1 2 (u i;j + u j;i + u 0 m;i u 0 m;j ) %Ov};|t CUOx@ @x @x 0 @x @y 0 @x @ 0 @y @x 0 @y @y 0 @x @ 0 @z @x 0 @z @y 0 @z @ 0 'TL pt=m Sv=Qo q |Ov@pwtQi R= xO=iDU= p}rOx@ 'OwW|t xOy=Wt xm Qw]u=ty |]N p}rLD QO xmv}= ut[ "CU= (X; Y ) |x}rw= Q=Di=@ x@ xHwD =@ G}=Dv s=tD w @N 0 i =@X "OQ=Ov OwHw @=@y =} @=@x w @=@Y =} @=@X u}@ |Dw=iD |x}L=v C=YDNt u}@ |x]@= Q u}vJty w OvDUy 20 |x]@= Q Ovv=ty @N 0 i =@Y CUOx@ 3 |xrO=at QO (x; y) uOQm u} Ro}=H =@ Ov=wD|t (X; Y ) w L i |ar [ [10]`H Qt R= xOWxDiQo |xrUt j}kO ?=wH w [21]`H Qt w 5 pmW "CU= <= RH= Q= Ri=sQv =@ |vwvm |xrUt '|O=yvW}B VwQ QDj}kO w QDy@ |xU}=kt |= Q@ xO=iDU= shell u=tr= R= Twm=@; |R=UpOt QO 'CU= xOW p}rLD R}v Twm=@; OwOLt 0 02 Q@=Q@ Approximate global size 'Mesh pwS=t QO |Ov@Vt |= Q@ w u; Technique w Tri u=tr= 'Element Shape CtUk QO "CU= xOW p=ta= p=ta= xrUt C=aq]= j@=]t C=YNWt Q}=U w RmQtDt |}=yDv= Q=@ w O=H}= Free QO p}rLD G}=Dv G= QNDU= w Twm=@; |xt=vQ@ \UwD xrUt p}rLD =@ "CU= xOW p}rLD R= pY=L G}=Dv w Twm=@; p}rLD R= pY=L G}=Dv u}@ |xU}=kt '7 pmW xO=O u=Wv |H=Hwa= w R} Q 'CWQO u=tr= 3 =@ ?rDt \}Lt QO |U} wvOm \UwD G}=Dv =@ |@ wN j@=]D '|O=yvW}B VwQ 'CU= XNWt xm Qw]u=ty "CU= xOW [11] " xrYw uwtR; |@=} RQ= |= Q@ K]Ut VvD |xrUt "4 pmW "xrYw uwtR; pL "1 pwOH ...
... As better described by Zienkiewicz and Taylor (2000), the Finite Element Method (FEM) is a numerical technique based on the subdivision of the domain into individual components, or 'elements', whose behavior is readily understood. These components are then assembled in a proper way to rebuild the original system in order to determine the approximate behavior of the body. ...
... The appropriate matrix N for each type of element can be easily found in the literature (e.g. Zienkiewicz and Taylor, 2000;Reddy, 2004;Logan, 2007;Weaver and Johnston, 1984). ...
The present work concerns the inclusion of functionalities into the free software INSANE - INteractive Structural ANalysis Environment for the physically non-linear thermal analysis in solid bodies by the Finite Element Method when the physical properties of the material are temperature dependent. INSANE is a multi-platform object-oriented computational system being developed at the Federal University of Minas Gerais - UFMG. Once the physical properties are temperature dependent, Newton's iterative method is used to calculate the desired solution. The implementation was tested comparing the numerical results to analytical solutions available in the literature. The segmentation and generalization of the INSANE's numerical core allowed the reuse of the existing classes to support the software expansion.
... the Zaremba-Jaumann objective stress rate, see [4,7,8], can be shown to be ...
... which is known as the material (Lagrangian) elasticity tensor, cf. [4,8]. It is conjugated to the second Piola-Kirchhoff stress tensor as . ...
The main goal of the article is to compare a computational efficiency of different implementation of a hyperelastic material model in the ABAQUS/Standard v. 6.14 [1]. The software offers basically two ways to proceed, namely UMAT and UHYPER user subroutines [2]. The procedures are employed to implement an isotropic, compressible neo-Hookean material model [3]. Corresponding stored-energy function,
constitutive equations and the consistent tangent operator are presented. These are essential to program the subroutines. Some theoretical and numerical aspects of different implementation approaches are discussed. As examples, a tube under axial compression and a contact problem of disc are considered. On the basis of obtained results, selected aspects of computational efficiency and quality of solutions are compared.
... subject to the finite element description (e.g. trusses, beams, plates, shells, solids) of the structure [33,31] under consideration (i.e. the digital twin/system [23,10]): ...
... The process of characterizing the vibro-acoustic behavior of complex built-up systems nowadays heavily relies on Computer Aided Engineering (CAE) tools in order to fulfill the time and budget demands of the industry. Among these tools, the Finite Element Method (FEM) [1] is widely known for modeling systems with high level of detail in a deterministic manner. However, as the frequency of interest increases, the behavior of the components becomes highly sensitive to changes in the properties or geometry and deterministic methods are no longer suitable. ...
Using Statistical Energy Analysis (SEA) to characterize the power flow within a vibroacoustic system is a challenging task when the subsystems have irregular shape and complex construction. Retrieving analytical solutions for the ordinary SEA parameters is nearly impractical without restricting simplifications and periodicity is usually not exploitable due to the lack of repetition patterns. A promising option to perform the power balance for such cases is to filter part of the information contained in a Finite Element Method (FEM) model of the system, in order to convert it into a SEA model. In this paper, the Lorentzian Frequency Average and the Nonparametric Random Matrix Theory are applied to randomize the dynamic stiffness matrix of the FEM components from a system of industrial application. The obtained direct field dynamic stiffness matrices are employed along the diffuse field reciprocity relationship as a general framework to determine the energetic content of each component. The results obtained with this procedure are evaluated against the ones from classical SEA and Monte Carlo techniques.
... Данная методика была ранее реализована в программном модуле Fidesys Composite отечественной CAE-системы Fidesys [6,19], предназначенном для оценки эффективных механических и теплофизических характеристик гетерогенных материалов (композиционных, пористых и т.п.). В статьях [3,11,13,14,15,16] описано использование Fidesys Composite для оценки эффективных свойств неоднородных материалов с помощью метода конечных элементов [17,18], а в статье [1] -с помощью более современного метода спектральных элементов [9]. Однако во всех этих статьях для оценки эффективных характеристик использовались только объёмные конечные/спектральные элементы. ...
Развитие аддитивных технологий (3D-печати) сделало возможным изготовление деталей и изделий регулярной пористой и ячеистой структуры (с целью облегчения конструкции). При этом характерный размер ячейки намного меньше масштаба целого изделия. Численные прочностные и смежные с ними расчёты подобных конструкций требуют предварительной оценки эффективных характеристик такой ячеистой структуры. В данной статье представлена методика численной оценки эффективных упругих характеристик регулярных ячеистых структур, основанная на численном решении краевых задач теории упругости на ячейке периодичности. К ячейке последовательно прикладываются различные периодические граничные условия в виде связей, наложенных на перемещения противоположных граней ячейки. Для каждого вида граничных условий решается краевая задача теории упругости, полученное в результате решения которой поле напряжений осредняется по объёму. Эффективные свойства ячеистого материала оцениваются в виде обобщённого закона Гука.
В работе рассматриваются композиционные материалы на основе жёсткого решётчатого каркаса, заполненного более мягким материалом. Расчёты проводятся методом конечных элементов с помощью отечественной CAE-системы «Фидесис». При этом в ряде расчётов для моделирования решётчатого каркаса используются конечные элементы балочного типа. В некоторых расчётах, помимо каркаса и матрицы, учитывается наличие тонкого слоя связующего между ними. Этот слой моделируется при помощи конечных элементов оболочечного типа.
Приводятся графики сравнения результатов расчётов композиционных материалов с решётчатым каркасом с моделированием каркаса балочными элементами и результатов аналогичных расчётов, в которых каркас моделируется трёхмерными конечными элементами. Также приводятся графики сравнения результатов расчётов, в которых слой связующего моделируется оболочечными элементами, с результатами аналогичных расчётов, в которых связующее моделируется трёхмерными элементами. Графики показывают, что при достаточно тонких элементах каркаса (либо при достаточно тонком слое связующего) результаты получаются довольно близкими, что подтверждает применимость балочных и оболочечных элементов для численного решения таких задач.
... (22)(23)(24)(25)(26) with the boundary conditions (27)(28)(29)(30)(31) are simulated numerically. Well description of this method can be found in the books and papers by Uddin et al. [28], Zienkiewicz et al. [41], Zienkiewicz et al. [42], Codina [43], Uddin [44], Rahman et al. [45] and Al Kalbani et al. [46]. Thus, the explanations of this method are absent here for shortness. ...
... We don't get into the details of the finite element method. For a consistent presentation of the method, the interested reader is referred to classical textbooks such as those of [ZT13,Hug00,Bat07]. ...
The Principle of Virtual Power (PVP) offers a systematic approach for studying the
equilibrium of complex systems. This chapter aims at showing the importance of the
principle and its use in discrete and continuum systems. Simple examples are given
throughout the chapter for helping understanding. The general theory of micromorphic
continua is also derived using the principle of virtual power. Finally, PVP being
global rather than local, is directly amenable to numerical schemes such as the Finite
Element Method (FEM). An example is given using the open-source FEM library
FEniCS.
... The numerical procedure used to solve the governing equations is based on the Galerkin weighted residual method of finite element formulation. Zienkiewicz and Taylor [10] has documented the applications of this technique. To solve the governing equations the non-linear parametric solution technique has been chosen, since it ensures substantially rapid convergence. ...
A numerical study of two-dimensional, laminar, steady mixed convection heat transfer in a Cu-water nanofluid filled lid-driven square cavity with an isothermally heated cylinder has been conducted. The wall of the cylinder is maintained at a constant high temperature, whereas the walls of the cavity (including the moving lid) are maintained at a constant low temperature. The isothermally heated cylinder is placed at the center of the cavity. The fluid flow in the cavity is driven by the combined effect of the buoyancy force due to temperature gradient and forced flow due to the top moving wall in the +x direction. The developed mathematical model is governed by the two-dimensional continuity, momentum and energy equations, which are solved by using Galerkin finite element method. The working fluid inside the cavity is Cu-water nanofluid, where water has been considered as the base fluid. The influence of the Reynolds number (1 ≤ Re ≤ 500) and the solid volume fraction of the Cu nanoparticle (0≤ ϕ ≤0.05) on fluid flow and heat transfer has been numerically investigated for the case of pure mixed convection heat transfer. Numerical results are presented in terms of the distribution of streamlines and isothermal contours, local as well as average Nusselt number variation on the cylinder surface for different parametric conditions. It is observed that enhancement of heat transfer occurs significantly as Reynolds number and solid volume fraction of nanoparticle change continuously. Thus, the dynamic condition of the moving lid and solid volume fraction of the nanoparticle can be used as parameters for enhancing the heat transfer characteristics and flow behavior in that cavity.
... [11]) at each step. The final value of FOS indicating the failure of the slope is determined according to the definition of failure proposed by Zienkiewicz and Taylor (2000). That is, after a specified number of iteration, no stress distribution can be found to satisfy both the local equilibrium, the Mohr-Coulomb criterion, and the global equilibrium (Eq. ...
Core Ideas
We examine SWCC and shear strength under various stress states and suction levels.
Relevant effective stress for granite‐weathered residual soil in Korea is proposed.
We propose a numerical framework for strength analysis with transient infiltration.
Potential failure mechanisms of civil infrastructure under rainfall are explored.
We suggest using stress‐dependent SWCC in strength analysis.
This study performed a series of experiments to examine the hydraulic‐mechanical properties of granite‐weathered residual soil in the Korean Peninsula. Particular attention was paid to the soil‐water characteristic curve (SWCC) and shear strength under various stress states and matric suction levels. The experimental results indicated the decisive influence of the stress state on the SWCC, notably in the low range of matric suction. In addition, the evolution of shear strength with suction became significant under high net confining stress. The effective stress using the stress‐independent SWCC could not describe the actual mechanical behaviors of the unsaturated soil. The relevant effective stress for the granite‐weathered residual soil in consideration was then proposed. Next, a numerical framework for strength analysis with infiltration was developed to manifest the practical applications of the experimental results. The analysis results revealed the potential failure mechanisms of the geotechnical infrastructures induced by rainfall. Ignoring the contribution of matric suction may lead to overly conservative outcomes and cannot capture the realistic performance of soil under the rainfall condition. Moreover, the stress‐dependent hydraulic properties are suggested for application in strength analysis for the safer design of geotechnical infrastructure.
... A coarsening algorithm is employed to reduce the number of finite elements. A penalty method [21][22][23][24][25] is used to solve the resulting linear system. We also provide a sensitivity analysis method to handle uncertainties in the estimation of the mechanical parameters (see Section 3.3.). Figure 13. ...
In surgical knee replacement, the damaged knee joint is replaced with artificial prostheses. An accurate clinical evaluation must be carried out before applying knee prosthe-ses to ensure optimal outcome from surgical operations and to reduce the probability of having long-term problems. Useful information can be inferred from estimates of the stress acting onto the bone-prosthesis system of the knee joint. This information can be exploited to tailor the prosthesis to the patient's anatomy. We present a compound system for pre-operative surgical planning based on structural simulation of the bone-prosthesis system, exploiting patient-specific data.
... The Sloan's profile reduction algorithm can be used to reduce the profile of symmetric sparse matrix stored in skyline form (variable band matrix storage). This form of storage is used by many FE programs, in particular, FEAP (Zienkiewicz & Taylor, 1998) in DLEARN (Hughes, 1987), MODEL programs (Akin, 1994). ...
This note provides some implementation details and tests of MATLAB implementation of the Sloan algorithm (Sloan, 1989) for profile and wavefront reduction. The program is freely available from
https://www.mathworks.com/matlabcentral/fileexchange/71934-reduceprofile
... The time integration of system (8) is carried out by the "Truncated Taylor series collocation algorithm GN22" (Zienkiewicz, 2000). The solution for the unknowns are parameters for time integration. ...
... The discrete models of variational equations (2)-(4), (12), (14)- (15) are built using the direct stiffness approach [22]. A computational region is divided in a finite number of nonoverlapping subregions. ...
... Many authors have used finite element method to predict accurate inplane stress distribution which is then used to solve the buckling problem. Zienkiewicz [43] and Cook [44] have clearly presented an approach for finding the buckling strength of plates by first solving the linear elastic problem for a reference load and then the eigenvalue problem for the smallest eigenvalue which then multiplied by the reference load gives the critical buckling load of the structure. An excellent review of the development of plate finite elements during the past 35 years was presented by Yang et al. [45]. ...
Finite element (FE) method is presented for the analysis of thin rectangular laminated composite decks plates under the biaxial action of in-plane compressive loading. The analysis uses the classical laminated plate theory (CLPT) which does not account for shear deformations. In this theory it is assumed that the laminate is in a state of plane stress, the individual lamina is linearly elastic, and there is perfect bonding between layers. The classical laminated plate theory (CLPT), which is an extension of the classical plate theory (CPT) assumes that normal to the mid-surface before deformation remains straight and normal to the mid-surface after deformation. Therefore, this theory is only adequate for buckling analysis of thin laminates. A Fortran program has been developed. The convergence and accuracy of the FE solutions for biaxial buckling of thin laminated rectangular plates was verified by comparison with various theoretical and experimental solutions. New numerical results are generated for in-plane compressive biaxial buckling which serve to quantify the effects of lamination scheme, aspect ratio, material anisotropy, fiber orientation of layers, reversed lamination scheme and boundary conditions. It was found that symmetric laminates are stiffer than the anti-symmetric one due to coupling between bending and stretching which decreases the buckling loads of symmetric laminates. The buckling load increases with increasing aspect ratio, and decreases with increase in modulus ratio. The buckling load will remain the same even when the lamination order is reversed. The buckling load increases with the mode number but at different rates depending on the type of end support. It is also observed that as the mode number increases, the plate needs additional support.
... Aby było możliwe utworzenie siatki elementów skończonych, należy wczytać siatkę złożoną z powierzchni i "wypełnić" objętością, a następnie w module Mesh wygenerować siatkę. W module Solver można wybrać format, do jakiego ma być wyeksportować plik [8]. ...
... If a proportional damping is used, it is possible to keep the same procedure and, at the end, the projected equations are still diagonal. Consider one particular type of proportional damping -see [126] for more general options-, the Rayleigh damping [41,136]: ...
This work fits within the Proper Generalized Decomposition (PGD) framework for the numerical solution of multidimensional partial differential equations. The computational performance of the PGD relies on the assumption that not only the solution but also the problem data admit a low-rank separated representation. This work is concerned with the separated formulation of the problem data. The solution to these separability issues is sought in two ways. The first strategy consists in looking for an efficient formulation such as to encompass the need of computing the separated representation. This is achieved by means of a frequency-based approach which allows overcoming the non-separability in the space-time domain of wave-like excitations. The efficiency of the approach is demonstated in both transient linear structural dynamics and heat transfer problems. Besides, the reciprocity principle can be proven thanks to the symmetrization introduced by the frequency-based formulation, yielding a direct application for the real-time monitoring of processes. The second strategy devises an innovative method to compute the separated representation of multivariate functions. An a priori interpolation-based approach is designed in order to effectively reduce the computational complexity associated to traditional projection-based methods. The performance of this method is demonstrated with application to non-separated and nonlinear coefficients.
... In addition, KUBC, SUBC and PERIODIC satisfy the Hill-Mandel macrohomogeneity condition [60,189 ], ensuring the micro-macro equivalence in terms of mechanical work density. In other words, the Hill-Mandel condition requires that the macroscopic volume average of the variation of work performed on the RVE is equal to the local variation of work on the macroscale and, if expressed in terms of the stress and strain quantities defined in Eq. (14), reads as < : >=< >:< >; (34) with ...
In a vast number of engineering fields like medicine, aerospace or robotics, materials are required to meet unusual performances that simple homogeneous materials are often not able to fulfil. Consequently, many efforts are currently devoted to develop future generations of materials with enhanced properties and unusual functionalities. In many instances, biological systems served as a source of inspiration, as in the case of cellular materials. Commonly observed in nature, cellular materials offer useful combinations of structural properties and low weight, yielding the possibility of coexistence of what used to be antagonistic physical properties within a single material. Due to their peculiar characteristics, they are very promising for engineering applications in a variety of industries including aerospace, automotive, marine and constructions. However, their use is conditional upon the development of appropriate constitutive models for revealing the complex relations between the microstructure's parameters and the macroscopic behavior. From this point of view, a great variety of analytical and numerical techniques have been proposed and exhaustively discussed in recent years. Noteworthy contributions, suggesting different assumptions and techniques are critically presented in this review paper. © 2018 The Authors. Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics
... For the FEM calculations, the ABAQUS/Standard program assumes that the loads are proportional, i.e. all load quantities differ only by a single scalar parameter. Because the shells are loaded via displacement boundary conditions, the standard Newton-Raphson method is used, which is the basic algorithm used for nonlinear boundary value problems [5][6][7]. ...
The aim of the article is to present the application of finite element method (FEM) programs ABAQUS/Standard [1] and ABAQUS/Explicit [2] and the constitutive models of incompressible isotropic hyperelastic materials [3] in the analysis of local and global buckling of axially compressed shell elements made of elastomers. Three FEM models of tubes with the same length and initial stiffness have been created for this purpose. These are tubes with elliptical, square and triangular cross-sections. Three types of constitutive models of a rubber-like (elastomeric) material are used - with the polynomial function of elastic energy in the form of the model MV [3] and standard models of Neo-Hooke and Mooney-Rivlin [4]. No imperfections are introduced in the FEM models of the analyzed pipes. Numerical simulations of buckling of pipes are performed for two types of initial-boundary value problems, i.e. quasi-static and dynamic ones. It has been shown that the type of buckling depends on the cross-section of the pipe. The solutions of buckling of pipes modelling with different constitutive models are compared and good correlations of the results have been observed.
... The optimum mesh density was determined based on a convergence study, this was achieved through consecutive simulations of a reference displacement (2mm), while altering the mesh density. Once two consecutive simulations result in a deviation of the monitored reaction force values of 0.2% [19], the grid was considered as 'mesh independent' and accepted for all further simulations. The final mesh consisted of 12.608 solid elements, with mean element side length of 2mm. ...
Determination of the mechanical behaviour of lattice structures has become a necessity to successfully implement lightweight concepts in various applications. Due to the large surface to volume ratio of these structures, the computational effort required for the FE simulation of models incorporating lattices is significantly increased mainly because of the large finite elements number which are necessary to accurately describe the complex input geometry. In this work a simple solution is employed in order to calculate the stress – strain properties, using a bilinear material law and equivalent bulk geometry. The verification of the FE model is fulfilled through the comparison of the lattice deformation and compressive force between experimental data and FE calculated results. After the verification of the FE model, it was possible to determine the mechanical behaviour of stochastic lattices, for an extended range of the investigated parameters, using only computational tools.
... Расчёты проведены в программном комплексе ANSYS APDL, использующем метод конечных элементов для дискретизации дифференциальных уравнений [24]. ...
This article contains experimental data about temperature fluctuations in the liquid metal flow near a channel wall of the model heat exchanger in conditions expected in the liquid metal tokamak-reactors. Methods for predicting the parameters of such fluctuations in real working conditions for various types of tokamak liquid metal cooling systems are proposed. Cyclic thermal stresses caused by the temperature fluctuations in the walls of the tokamak heat transfer system under real operating conditions were estimated with the goal to evaluate, in the first approximation, potential risk of such abnormal fluctuations. © 2018 National Research Center Kurchatov Institute. All Rights Reserved.
... where P, C11, and v are the pressures which are equal to mean stresses, the bulk modulus and the volumetric strains of the fluid, respectively. Since irrotational behavior of the fluid is considered like penalty methods [10,11], rotations and constraint parameters are included in the pressure-volumetric strain equation (Eq. (1)) of the fluid. ...
The aim of this paper is to present accelerations and peak acceleration amplification coefficients in a concrete faced rockfill (CFR) dam. For this purpose, Torul CFR dam is considered by using finite element method. Hydrodynamic pressure of the reservoir water is considered modeled using the fluid finite elements based on the Lagrangian approach. Deconvolution of surface accelerograms is taken into account in numerical solutions. Geometric, material and connection non-linearity are considered in the finite element analyses. The Drucker-Prager model and the multi-linear kinematic hardening model are used for concrete slab and rockfill, respectively, in the materially non-linear analysis. Non-linear behavior of the rockfill is obtained by the uniaxial stress-strain relation. The stress-strain curve of the rockfill is obtained using the shear modulus-shear strain relation produced for the gravels. Different joints in the CFR dam are modeled with both welded and friction contacts. The friction in the joints is provided with contact-target element pairs based on the Coulomb’s friction law. The accelerations in the bottom and crest of the dam are compared each other since welded and friction contact in dam-foundation interface. Then peak acceleration amplification coefficients are also obtained compared each other. According to numerical comparisons, when the rockfill behaves materially non-linear, the maximum horizontal accelerations and peak acceleration amplification coefficients decrease in the crest.
... The FEM has been used to solve the discrete motion free equation, Zienkiewicz and Taylor (2000): ...
Shared link to full paper: https://rdcu.be/1ZzI
Suspended glass panels are monolithic or laminated frameless windows sustained by a number of holders, typically located in the vicinity of the edges. These panels can be used, among other purposes, as noise barriers. The vibro-acoustic behaviour of glass windows is critical at low frequencies, where the problem is often tackled by increasing the thickness, thus the mass,of the panels. As a consequence, solutions which preserve low mass are greatly sought by industries. In this study, the vibro-acoustic behaviour of different suspended glass panels is addressed. An optimization procedure is implemented, aiming at finding the position of the holders which maximizes the acoustic transmission loss (TL) averaged at low- and very low-frequency ranges. First, an iterative procedure, based on comparison of experimental and numerical modal data, has been implemented to extract the material properties (Young’s modulus and Poisson’s ratio) of the panels. Second, these properties have been used in an optimization procedure based on finite difference approximation of the objective function, the averaged transmission loss. The vibro-acoustic analyses, required by the optimization procedure, have performed by means of hybrid finite element method/statistical energy analysis (FEM/ SEA). 16 different design cases have been considered in the optimizations, i.e. 2 different frequency ranges (20-300 Hz and 20-1000 Hz), 2 panel geometries (square 1m x 1m and rectangular 2.5m x 0.8m), 2 constitutive material properties (monolithic tempered glass and laminated tempered glass) and 2 mounting solutions (4 and 6 holders). The transmission losses of the optimized and the standard configurations, where the holders are placed close to the edges, are compared.
... This results in a fully-predictive approach in which both hydraulic and mechanical processes are simulated by using the same WRC. For this purpose, here the Richards equation ( Richards, 1931) is solved through a Galerkin finite element scheme ( Zienkiewicz et al., 1977) to simulate the transient variation of suction upon infiltration: ...
The stability of unsaturated slopes is inherently linked to the prevailing hydrological and mechanical states. Their combination with unsaturated soil properties can lead to different slope failure mechanisms upon infiltration, thus having important implications on landslide hazard studies. This paper uses plasticity theories for unsaturated soils to formulate safety factors for shallow slopes. The objective is to define initiation conditions for frictional slips and liquefaction-induced flowslides that allow their simultaneous assessment as a function of the hydromechanical variables and initial conditions. The expressions are derived with reference to the kinematics of infinite slopes, as well as by detecting constitutive singularities of a suction-dependent constitutive model. The theory has been combined with a finite element solver able to simulate transient infiltration in unsaturated porous media, thus testing it against data from flume tests on collapsible soils. The simulations show that the model is capable of capturing the variation of the failure modes resulting from different combinations of initial suction and density. These results support the use of advanced mechanical theories to evaluate the susceptibility to rainfall-induced landslides, by encompassing frictional slips and flowslides within the same framework.
... Time discretisation is achieved by means of the standard Finite Difference Method. A non-symmetric, nonlinear system is finally obtained, and linearisation, usually by means of the Newton-Raphson method, is required (Zienkiewicz and Taylor 2000). ...
This chapter deals with the problem of modelling the behaviour of massive concrete structures. In the last decades, the developments in the field of computational mechanics were significant, so nowadays several numerical techniques are available to this goal, depending on the scale level considered but also on which phenomena/processes are taken into account. In this chapter, we limit the description to approaches/models that can be implemented using the Finite Element Method, which is still the worldwide most used numerical technique. This chapter presents two distinct groups of models. The first group covers deterministic models starting from the simplest ones, which consider simply the thermo-chemo-mechanical behaviour of the material, to more sophisticated approaches which consider also the fluid phases; i.e., they consider concrete as a multiphase porous material. In this first part, a specific section is dedicated to mechanical behaviour modelling considering damage of the material, plasticity, etc. The second group of the models takes into consideration stochastic nature of cracking. These models are formulated specifically for giving a detailed information about cracks spacing and opening in concrete structures in service life conditions.
... This principle will be briefly described. For more details see Zienkiewicz [46] and Bathe [47]. The virtual work principle states that the equilibrium of a body requires that for any compatible small virtual displacements imposed on the body in its state of equilibrium, the total internal virtual work is equal to the total external virtual work: ...
A structural intervention proposal is presented for a segment of a twentieth century building that presents damages due to a drain of graded materials built in an earlier retrofit; the damage increased when the building was subjected by the September 19, 2017 earthquake. Currently, the building is used as a clubhouse in a well-known residential area located in Tarímbaro, Michoacán, México. The structure is composed of irregular stonework walls joined with mud mortar. The building’s original cover roof consisted of wooden beams, lower shingle, terrace and a brick upper cover that was replaced in the 1970’s by a solid reinforced concrete (RC) slab. RC slab provided a high stiff horizontal diaphragm that could have contributed to the increase in wall cracking during the 2017 earthquake, due to the incompatibility of both materials. On the other hand, it provided a beneficial effect by keeping tied the walls damaged by the severe ground settlement generated in the eastern façade, caused by water seepage into the drain built without an adequate exit to the existing original drainage system in the construction. Based on the recommendations of the International Scientific Committee ISCARSAH of ISCARSAH (International Scientific Committee for Analysis and Restoration of Structures of Architectural Heritage. Recommendations for the Analysis, Conservation and Structural Restoration of Architectural Heritage - ICOMOS, France. https://iscarsah.files.wordpress.com/2014/11/iscarsah-guidelines.pdf, 2003), a qualitative and quantitative diagnosis of the structure is presented using a methodology based on a mesh of finite elements with linear behavior. The results showed that current damages appreciably reduced the building safety, so a retrofit proposal for the foundation and walls is presented, which was carried out successfully.
The Method of Weighted Residuals (MWR) is used to compare the finite element method (FEM) with the finite volume method (FVM) through nodal recursion relations. Both methods reside under the general MWR structure, with the underlying switch between the two methods established through the weighting function. Both methods yield comparable spatial accuracy for steady-state conditions. However, the flexibility of the FEM permits additional options that can increase accuracy, but generally at the expense of additional time and resource constraints.
In Chap. 10.1007/978-3-030-37685-7_4, an overview of three ABS-designed prototypes has been presented and their main features have been discussed. In particular, after the design of a first tentative mechanical solution which allowed for the kinematic chain validation (Sect. 10.1007/978-3-030-37685-7_4) and the first development of this device embodied by a new system tailored on the user’s hand (Sect. 10.1007/978-3-030-37685-7_4), a fully wearable and portable hand exoskeleton has been reached with the third prototype (Sect. 10.1007/978-3-030-37685-7_4).
Dry stone retaining walls (DSRWs) are vernacular structures, which consist in a specific assemblage of individual rubble stones. Herein, we propose some recommendations to achieve a correct modelling of the mechanical behaviour of DSRWs towards failure with the use of the Discrete Element Method (DEM). There are two kinds of DSRWs, those that just retain slope and others that retain a road built on the top of the backfill. For walls retaining slopes, a full 2D DEM approach (PFC2D) was used as the most sophisticated way to study such a system and the modelling was validated on full scale experiments. The modelling retrieved in a very good way the features observed on site as the expense of much computation time. More interesting was the use of a mixed DEM-continuum approach (UDEC) where the constitutive laws for the stone-stone contact and backfill-wall contact are averaged. Very efficient in terms of computation time, the process of identification of the model parameters is here much lighter than with a full DEM approach without losing precision for the prediction. For walls retaining a slope with a highway a mixed 3D DEM-continuum approach (3DEC) was required. A straightforward methodology was proposed in order to correctly retrieve the features observed in both scaled down experiments or in full scale experiments. In conclusion, the full DEM and the mixed DEM-continuum approaches were able to precisely capture the mechanical behaviour of DSRWs towards failure even if idealised blocks were used.
Piston diaphragm pumps are used worldwide to transport abrasive and aggressive slurries against high discharge pressures in the mining, mineral processing, and power industries. Intermittent suction and drainage of the diaphragm, however, can lead to pulsating output pressure, which has caused major problems in the application of these pumps. To improve the accuracy of simulations of piston diaphragm pumps and to enable better simulation of their pressure pulsation behavior, it is necessary to carry out a three-dimensional fluid–structure interaction simulation of the pump fluctuation characteristics. This article proposes a simplified simulation model based on the periodic motion characteristics of a piston diaphragm pump, where a ZMB240 piston diaphragm pump serves as the research object. By simplifying the numerical simulation of the model, we are able to analyze the deformation characteristics and the fluctuation characteristics of the piston diaphragm pump under different initial conditions for the pressure-stabilizing air chamber.
Thin and moderately thick shell structures are designed as structural components in many engineering applications because of light weight and high load-carrying capacity. In many cases they are subjected to high temperature environment and mechanical loadings, such that inelastic material behavior must be taken into account. Examples of high-temperature shell components include pressure vessels, boiler tubes, steam transfer lines, thin coatings, etc. A steam transfer line under long-term operation considering creep-damage material behavior is discussed in Naumenko and Altenbach (2016, Chap. 1). Chapter 5 presents examples of inelastic structural analysis of plates and shells. Section 5.1 gives an overview of modeling approaches including various theories of plates and shells as well as various constitutive models of inelastic material behavior. Governing equations of the first order shear deformation theory of plates are presented in Sect. 5.2. An emphasis is placed on the direct formulation of inelastic constitutive laws. Section 5.3 illustrates examples of steady-state creep analysis of circular plates. Advanced constitutive models with internal state variables, such as the damage parameter require the use of advanced plate theories to consider edge effects. Section 5.4 illustrates an example of a rectangular plate with different types of boundary conditions. The results based on the plate theory are compared with the results according to the three-dimensional theory. Section 5.5 presents governing equations and the solution procedure for the creep behavior of a thin-walled pipe subjected to the internal pressure and the bending moment.
Elastic guided wave based evaluation is useful to detect delamination in composites. This paper reports a detailed account of guided wave interaction with delamination in laminated composites modeled using time domain spectral finite element (TSFE) method. Wave scattering due to different delamination positions is studied. Wave field near the delamination in different layer-wise positions is investigated, which is useful to further characterize and correlate the far field wave packet carrying the parametric signature of delamination. Off-axis delamination in composite laminate creates an asymmetry in the elastodynamic stress transfer. It leads to wave mode conversions. Sensitivity of the delamination to the wavelengths is studied. The outcome of this study indicates potential possibilities to use frequency-wavelength information to discriminate various sizes of delamination. Energy of the scattered waves and dissipation/conversion of the wave energy due to the defects depend on the resonance characteristics of the sub-laminates. Wave scattering effect due to length-wise multiple delaminations and edge delamination is also analyzed. The resonance patterns in the signals are analyzed with reference to the defect quantification problem.
This paper deals with the probability and sensitivity analysis of the foundation slab rested on the layered elastic half-space. The slab has been modeled with shell elements SHEEL185, solid earth masses with solid elements Solid185 with assignment of material properties to individual substrate layers. The CONTA52 was used as a contact element in the ANSYS software. Probability and sensitivity analysis was performed using approximation RSM method (Response Surface Method) for maximum allowable deflection according to the second limiting state. The individual input parameters was varied according to Gaussian, Uniformly or Long-normal probability distribution.
In the paper, a method of automatic choice of the boundary conditions is presented. This method allows the initial estimation the value of the boundary conditions during the heating operation for the welding process. The solutions described in this paper can successfully support the work of the specialists performing numerical simulations. They also significantly reduce the time required to estimate the boundary conditions parameters consistent with the actual experiment. The accuracy of the solution in the presented method is not dependent on the empirical models of boundary conditions for the heating process. It is also much more universal. Artificial neural networks after the learning process can be used to perform a reverse analysis for the heat treatment process, for example, to determine the parameters of welding process in the production process of steel elements.
В книге излагаются основы метода конечных элементов применительно к нелинейным задачам. Приводятся соотношения для стандартных конечных элементов, включая плоские, осесимметричные, объемные, стержневые, плитные и оболочечные. Формулируется целый класс нестандартных элементов, характерных для нелинейных задач. Анализируются нелинейные модели таких материалов, как бетон и железобетон, грунты, как в виде ассоциированных и неассоциированных теорий течения, так и в форме деформационных теорий. Дается подробное описание программного комплекса, ориентированного на решение нелинейных задач методом конечных элементов с использованием шагово-итерационных процедур. Рассматриваются не только вычислительные алгоритмы, положенные в основу комплекса, но также и алгоритмы, используемые в сервисной части и предназначенные как для графического ввода исходных данных, так и для визуализации результатов расчета. Большая часть теоретического материала сопровождается численными алгоритмами в виде исходных текстов программ на Фортране и Си. Приводятся наглядные примеры решения нелинейных задач из самых разнообразных технических и геотехнических приложений.
Книга предназначена для специалистов, занимающихся расчетами и проектированием сооружений, а также для преподавателей, аспирантов и студентов вузов строительных и гидротехнических специальностей.
A novel regularized interface integral equation for three-dimensional steady state heat conduction problems with non-homogeneous inclusions is developed. The proposed formulation only contains the fundamental solution of isotropic matrix. As a result, the fundamental solution of non-homogeneous inclusion, usually very difficult to obtain, is avoided. Domain integrals caused by the contrast of heat conductivities between the inclusions and the matrix are converted into equivalent interface integrals using the radial integration method by expressing the temperature gradient as a series of radial basis functions. Therefore, a pure interface integral equation is obtained and there is no need to discretize the inclusion into finite elements to evaluate the domain integral. For the determination of the flux and temperature, collocation points are distributed inside the inclusion to form a system of linear equations. To eliminate the geometrical errors and study the inclusions with arbitrary geometry, bivariate Non-Uniform Rational B-Splines basis functions are used to depict the boundaries of the inclusions. Numerical results are compared with available analytical solutions or finite element solutions.
The migration of contaminant through soil is usually modeled using the advection‐dispersion equation and assumes that the porous media is stationary without introducing a constitutive equation to represent soil structure. Consequently, time‐dependent deformation induced by soil consolidation or physical remediation is not considered, despite the need to consider these variables during planning for the remediation of contaminated ground, the prediction of contaminated groundwater movement, and the design of engineered landfills. This study focuses on the numerical modeling of solute transfer during consolidation as a first step to resolve some of these issues. We combine a coupling theory‐based mass conservation law for soil‐fluid‐solute phases with finite element modeling to simulate solute transfer during deformation and groundwater convection. We also assessed the sensitivity of solute transfer to the initial boundary conditions. The modeling shows the migration of solute toward the ground surface as a result of ground settlement and the dissipation of excess pore water pressure. The form of solute transport is dependent on the ground conditions, including factors such as the loading schedule, contamination depth, and water content. The results indicate that an understanding of the interaction between coupling phases is essential in predicting solute transfer in ground deformation and could provide an appropriate approach to ground management for soil remediation.
This chapter focuses on rectangular elements solved by displacement. It presents the grid and calculated results of the analysis of shear wall lateral displacement of six‐story buildings using 4‐node rectangular elements. The rectangular elements are adopted in regular parts and the triangular elements in irregular parts. However, when adopting high‐order elements, the number of elements is always small and the sizes of elements are always large. As for high‐order triangular element, if still using orthogonal coordinates to define the shape function, the formula about the stiffness matrix will be very complicated. When the displacement functions are represented by Cartesian coordinates, the calculations of coefficients β1‐β12 and the stiffness matrix and nodal loads are very tedious, and the use of area coordinates can make it greatly simplified.
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