Oliver Rheinbach

Technische Universität Bergakademie Freiberg · Faculty of Mathematics and Computer Science and University Computing Center (URZ)

A computational framework is presented to numerically simulate the effects of antihypertensive drugs, in particular calcium channel blockers, on the mechanical response of arterial walls. A stretch-dependent smooth muscle model by Uhlmann and Balzani is modified to describe the interaction of pharmacological drugs and the inhibition of smooth muscl...

Multilevel extensions of overlapping Schwarz domain decomposition preconditioners of Generalized Dryja–Smith–Widlund (GDSW) type are considered in this paper. The original GDSW preconditioner is a two-level overlapping Schwarz domain decomposition preconditioner, which can be constructed algebraically from the fully assembled stiffness matrix. The...

A computational framework is presented to numerically simulate the effects of antihypertensive drugs, in particular calcium channel blockers, on the mechanical response of arterial walls. A stretch-dependent smooth muscle model by Uhlmann and Balzani is modified to describe the interaction of pharmacological drugs and the inhibition of smooth muscl...

Monolithic fluid–structure interaction (FSI) of blood flow with arterial walls is considered, making use of sophisticated nonlinear wall models. These incorporate the effects of almost incompressibility as well as of the anisotropy caused by embedded collagen fibers. In the literature, relatively simple structural models such as Neo-Hooke are often...

Numerical simulation of the response of healthy and pathological arteries to cardiovascular agents can provide valuable information to the physician in the treatment of diseases such as hypertension, atherosclerosis, and the Marfan syndrome. Here, we provide a first step towards a computational framework to model the effects of antihypertensive age...

We consider adaptive finite elements, using the open source finite element library deal.II [1], and an implementation [11] of the FETI-DP (Finite Element Tearing and Interconnecting Dual–Primal) method based on PETSc, for the solution of problems from dislocation micromechanics. The library deal.II is well known for its adaptive finite element appr...

Nonlinear FETI-DP (Finite Element Tearing and Interconnection - Dual Primal) methods [10] are nonlinear generalizations of linear FETI-DP domain decomposition methods [5, 16]. Nonlinear FETI-DP domain decomposition methods have shown their robustness and scalability, e.g., for linear and nonlinear structural mechanics problems [11], where results f...

The Fast and Robust Overlapping Schwarz framework [7, 8], which is part of the Trilinos Software library [18], contains a parallel implementation of the generalized Dryja–Smith–Widlund (GDSW) preconditioner.

We consider the swelling of hydrogels as an example of a chemo-mechanical problem with strong coupling between the mechanical balance relations and the mass diffusion. The problem is cast into a minimization formulation using a time-explicit approach for the dependency of the dissipation potential on the deformation and the swelling volume fraction...

We consider the swelling of hydrogels as an example of a chemo-mechanical problem with strong coupling between the mechanical balance relations and the mass diffusion. The problem is cast into a minimization formulation using a time-explicit approach for the dependency of the dissipation potential on the deformation and the swelling volume fraction...

We construct two-dimensional, two-phase random heterogeneous microstructures by stochastic simulation using the planar Boolean model, which is a random collection of overlapping grains. The structures obtained are discretized using finite elements. A heterogeneous Neo-Hooke law is assumed for the phases of the microstructure, and tension tests are...

The globalization of Nonlinear FETI-DP (Dual Primal Finite Element Tearing and Interconnecting) methods is considered using a Sequential Quadratic Programming (SQP) approach. Nonlinear FETI-DP methods are parallel iterative solution methods for nonlinear finite element problems, based on divide and conquer, using Lagrange multipliers. In these meth...

Our general goal is to study chemo-mechanical problems at various length scales by means of a fully-integrated approach in terms of a co-design of variational formulations and tailored parallel solvers. Based on prior experience in electro-magneto-mechanics [1], we know that the advantages and disadvantages of different variational settings for suc...

Different graph partitioning methods, i.e., linear partioning, parallel hypergraph (PHG) partioning, and two approaches using ParMETIS, are considered to generate an unstructured decomposition of the second-level coarse operator of three-level FROSch (Fast and Robust Overlapping Schwarz) preconditioners in the Trilinos software library. In our cont...

The Nakajima test is a well-known material test from the steel and metal industry to determine the forming limit of sheet metal. It is demonstrated how FE2TI, our highly parallel scalable implementation of the computational homogenization method FE $$^2$$ 2 , can be used for the simulation of the Nakajima test. In this test, a sample sheet geometry...

Regression or regression-like models are often employed in mineral prospectivity modeling, i.e., for the targeting of resources, either based on 2D map images or 3D geomodels both in raster mode or based on spatial point processes. Machine learning techniques like artificial neural networks are often applied and give decent results in the predictio...

The parallel performance of the three-level Fast and Robust Overlapping Schwarz (FROSch) preconditioners is investigated for linear elasticity. The FROSch framework is part of the Trilinos software library and contains a parallel implementation of different preconditioners with energy minimizing coarse spaces of GDSW (Gen-eralized Dryja-Smith-Widlu...

This study represents a first step towards tailored solvers for chemo‐mechanical multi‐field problems in a variational setting.

Adaptive coarse spaces for domain decomposition methods are an active area of research to make iterative domain decomposition methods robust with respect to large discontinuities in the material parameters or almost incompressible elasticity. In order to make their use feasible for applications, the computational overhead of the adaptive methods ha...

In order to obtain a scalable domain decomposition method (DDM) for elliptic problems, a coarse space is necessary and an associated coarse problem has to be solved in each iteration. In the presence of arbitrary, large coefficient jumps or in case of almost incompressible elastic materials, the convergence rate of standard DDM deteriorates.

The GDSW (Generalized Dryja–Smith–Widlund) preconditioner is a two-level overlapping Schwarz domain decomposition preconditioner [23] with exact local solvers [5, 4].

This article describes a parallel implementation of a two-level overlapping Schwarz preconditioner with the GDSW (Generalized Dryja–Smith–Widlund) coarse space described in previous work [12, 10, 15] into the Trilinos framework; cf. [16]. The software is a significant improvement of a previous implementation [12]; see Sec. 4 for results on the impr...

A nonlinear domain decomposition (DD) solver is considered with respect to improved energy efficiency. In this method, nonlinear problems are solved using Newton’s method on the subdomains in parallel and in asynchronous iterations. The method is compared to the more standard Newton-Krylov approach, where a linear domain decomposition solver is app...

A new reduced dimension adaptive GDSW (Generalized Dryja-Smith-Widlund) overlapping Schwarz method for linear second-order elliptic problems in three dimensions is introduced. It is robust with respect to large contrasts of the coefficients of the partial differential equations. The condition number bound of the new method is shown to be independen...

Regression models are often employed in prospectivity modeling for the targeting of resources. Logistic regression has a well understood statistical foundation and uses an explicit model from which knowledge can be gained about the underlying phenomenon. In this paper, a model selection procedure based on logistic regression enhanced with nonlinear...

We present a numerical two-scale simulation approach of the Nakajima test for dual-phase steel using the software package FE2TI, a highly scalable implementation of the well known homogenization method FE2. We consider the incorporation of contact constraints using the penalty method as well as the sample sheet geometries and adequate boundary cond...

A parallel FETI-DP domain decomposition method using an adaptive coarse space is presented. The implementation builds on a recently introduced adaptive FETI-DP approach for elliptic problems in three dimensions and uses small, local eigenvalue problems for faces and, additionally, for a small number of edges. The condition number of the preconditio...

Regression models are often employed in prospectivity modeling for the targeting of resources. Logistic regression has a well understood statistical foundation and uses an explicit model from which knowledge can be gained about the underlying phenomenon. In this paper, a model selection procedure based on logistic regression enhanced with nonlinear...

In FETI-DP (Finite Element Tearing and Interconnecting) and BDDC (Balancing Domain Decomposition by Constraints) domain decomposition methods, the convergence behavior of the iterative scheme can be improved by implementing a coarse space using a transformation of basis and local assembly. This is an alternative to coarse spaces implemented by defl...

Parallel computational homogenization using the well-knwon \(\hbox {FE}^2\) approach is described and combined with domain decomposition and algebraic multigrid solvers. It is the purpose of this paper to show that and how the \(\hbox {FE}^2\) method can take advantage of the largest supercomputers available and those of the upcoming exascale era f...

Regression or regression‐like models are often employed in potential modeling, i.e., for the targeting of resources, either based on 2D map images or 3D geomodels both in raster mode or based on spatial point processes. Recently, machine learning techniques such as artificial neural networks have gained popularity also in potential modeling. Using...

FETI‐DP (Finite Element Tearing and Interconnecting Dual‐Primal) solvers and the deal.II adaptive finite element library are combined to solve dislocation eigenstrain problems in micromechanics. Computational results using adaptive finite elements with millions of unknowns and up to 3072 cores of the Taurus supercomputer at ZIH in Dresden are prese...

Parallel computational results for problems in dislocation mechanics are presented using the deal.II adaptive finite element software and the Fast and Robust Overlapping Schwarz (FROSch) Preconditioner.

A robust two-level overlapping Schwarz method for scalar elliptic model problems with highly varying coefficient functions is introduced. While the convergence of standard coarse spaces may depend strongly on the contrast of the coefficient function, the condition number bound of the new method is independent of the coefficient function. Indeed, th...

A three-level extension of the GDSW overlapping Schwarz preconditioner in two dimensions is presented, constructed by recursively applying the GDSW preconditioner to the coarse problem. Numerical results, obtained for a parallel implementation using the Trilinos software library, are presented for up to 90,000 cores of the JUQUEEN supercomputer. Th...

In this contribution, Fluid‐Structure‐Interaction (FSI) in blood vessels, in detail the simulation of realistic arterial geometries, where the interaction of the blood flow and the vessel wall is of special interest, is considered. Based on pervious research, cf. [1], our existing framework for FSI‐simulations is extended towards realistic arterial...

Adaptive coarse spaces for domain decomposition methods are an active area of research to make iterative domain decomposition methods robust with respect to large discontinuities in the material parameters or almost incompressible elasticity. In order to make their use feasible for applications, the computational overhead of the adaptive methods ha...

In FETI-DP (Finite Element Tearing and Interconnecting) and BDDC (Balancing Domain Decomposition by Constraints) domain decomposition methods, the transformation-of-basis approach is used to improve the convergence by combining the local assembly with a change of basis. Suitable basis vectors can be constructed by the recently introduced adaptive c...

Two-level overlapping Schwarz domain decomposition methods for second-order elliptic problems in two dimensions are proposed using coarse spaces constructed from the Approximate Component Mode Synthesis (ACMS) multiscale discretization approach. These coarse spaces are based on eigenvalue problems using Schur complements on subdomain edges. It is t...

New nonlinear BDDC (Balancing Domain Decomposition by Constraints) domain decomposition methods using inexact solvers for the subdomains and the coarse problem are proposed. In nonlinear domain decomposition methods, the nonlinear problem is decomposed before linearization to improve concurrency and robustness. For linear problems, the new methods...

We consider a recent overlapping Schwarz method with an energy-minimizing coarse space of reduced size. In numerical experiments for up to 64,000 cores, we show that the parallel efficiency and the total time to solution is improved significantly, compared to our previous overlapping Schwarz method using an alternative energy-minimizing coarse spac...

A highly scalable implementation of an inexact BDDC (Balancing Domain Decomposition by Constraints) method is presented, and scalability results for linear elasticity problems in two and three dimensions for up to 131,072 computational cores of the JUQUEEN BG/Q are shown. In this method, the inverse action of the partially coupled stiffness matrix...

Adaptive FETI-DP and BDDC methods are robust methods that can be used for highly heterogeneous problems when standard approaches fail. In these approaches, local generalized eigenvalue problems are solved approximately, and the eigenvectors are used to enhance the coarse problem. Here, a few iterations of an approximate eigensolver are usually suff...

We introduce an energy minimizing nonlinear preconditioner for our nonlinear FETI-DP methods, and we will show numerical results for some problems in two dimensions based on the scaled p-Laplace operator. The equivalence of nonlinear FETI-DP methods and specific right-preconditioned Newton-Krylov methods was already shown. In nonlinear FETI-DP meth...

We propose robust coarse spaces for two-level overlapping Schwarz preconditioners, which are extensions of the energy minimizing coarse space known as GDSW (Generalized Dryja, Smith, Widlund). The resulting two-level methods with adaptive coarse spaces are robust for second order elliptic problems in two dimensions, even in presence of a highly het...

In this contribution, results regarding fluid-structure interaction (FSI) simulations for three-dimensional arterial walls are presented. In detail, a benchmark problem for FSI simulations in arteries of sufficient complexity, which combines sophisticated nonlinear models for the fluid and the structure, cf. [1], as well as a short segment from a p...

Parallel Newton--Krylov FETI-DP (Finite Element Tearing and Interconnecting---Dual-Primal) domain decomposition methods are fast and robust solvers, e.g., for nonlinear implicit problems in structural mechanics. In these methods, the nonlinear problem is first linearized and then decomposed into loosely coupled (linear) problems, which can be solve...

Axel Klawonn, Martin Kühn, and Oliver Rheinbach. Adaptive FETI-DP and BDDC methods with a generalized transformation of basis for heterogeneous problems. Electron. Trans. Numer. Anal., 49:1–27, 2018. DOI: 10.1553/etna_vol49s1 @article{etna_vol49_pp1-27, author = {Axel Klawonn and Martin K\"uhn and Oliver Rheinbach}, title = {Adaptive FETI-DP and...

http://tu-freiberg.de/sites/default/files/media/fakultaet-fuer-mathematik-und-informatik-fakultaet-1-9277/prep/2017-01_fertig.pdf

Parallel results obtained with a new implementation of an overlapping Schwarz method using an energy minimizing coarse space are presented. We consider structured and unstructured domain decompositions for scalar elliptic and linear elasticity model problems in two dimensions. In particular, strong and weak parallel scalability studies for up to 10...

A new adaptive coarse space approach including a condition number bound for FETI-DP or BDDC methods for problems with coefficient jumps inside subdomains and across subdomain boundaries in three dimensions is presented. The approach is based on a known adaptive coarse space approach enriched by a small number of additional local edge eigenvalue pro...

We introduce two new nonlinear FETI-DP (Finite Element Tearing and Interconnecting—Dual-Primal) methods based on a partial nonlinear elimination of variables and provide a comparison to Newton-Krylov-FETI-DP, Nonlinear-FETI-DP-1, and Nonlinear-FETI-DP-2, which have already been described earlier. In contrast to classical Newton-Krylov-FETI-DP metho...

A Newton-Krylov-FETI-DP method for solving nonlinear partial differential equations is presented. The FETI-DP method, which is applied in each Newton step, has an adaptively enriched coarse space to deal with ill-conditioned linearized operators. The adaptive coarse spaces are obtained by solving local generalized eigenvalue problems. Heuristic str...

We describe a new implementation of a two-level overlapping Schwarz preconditioner with energy-minimizing coarse space (GDSW: generalized Dryja--Smith--Widlund) and show numerical results for an additive and a hybrid additive-multiplicative version. Our parallel implementation makes use of the Trilinos software library and provides a framework for...

A variant of a nonlinear FETI-DP domain decomposition method is considered. It is combined with a parallel algebraic multigrid method (BoomerAMG) in a way which completely removes sparse direct solvers from the algorithm. Scalability to 524,288 MPI ranks is shown for linear elasticity and nonlinear hyperelasticity using more than half of the JUQUEE...

It has been said that the story of materials is the story of civilization. However, it is clear that, throughout the history of civilization, from the iron age to modern days, iron and steel have been among the most versatile materials known by humanity. Through processing, the mechanical properties of steel can be controlled over a very wide range...

For second-order elliptic partial differential equations large discontinuities in the coefficients yield ill-conditioned stiffness matrices. The convergence of domain decomposition methods (DDM) can be improved by incorporating (numerically computed) local eigenvectors into the coarse space. Different adaptive coarse spaces for DDM have been constr...

An adaptive coarse space approach including a condition number bound for dual primal finite element tearing and interconnecting (FETI-DP) methods applied to three dimensional problems with coefficient jumps inside subdomains and across subdomain boundaries is presented. The approach is based on a known adaptive coarse space approach enriched by a s...

The convergence rate of iterative substructuring methods generally deteriorates when large discontinuities occur in the coefficients of the partial differential equations to be solved. In dual-primal Finite Element Tearing and Interconnecting (FETI-DP) and Balancing Domain Decomposition by Constraints (BDDC) methods, sophisticated scalings, e.g., d...

The parallel performance of several classical Algebraic Multigrid (AMG) methods applied to linear elasticity problems is investigated. These methods include standard AMG approaches for systems of partial differential equations such as the unknown and hybrid approaches, as well as the more recent global matrix (GM) and local neighborhood (LN) approa...

In this paper, aspects of the two-scale simulation of dual-phase steels are considered. First, we present two-scale simulations applying a top-down one-way coupling to a full thermo-elastoplastic model in order to study the emerging temperature field. We find that, for our purposes, the consideration of thermo-mechanics at the microscale is not nec...

Parallel overlapping Schwarz preconditioners are considered and applied to the structural block in monolithic fluid-structure interaction (FSI). The two-level overlapping Schwarz method uses a coarse level based on energy minimizing functions. Linear elastic as well as nonlinear, anisotropic hyperelastic structural models are considered in an FSI p...

A new nonlinear version of the well-known FETI-DP method (Finite Element Tearing and Interconnecting Dual-Primal) is introduced. In this method, the nonlinear problem is decomposed before linearization. Nonlinear approaches to domain decomposition can be viewed as a strategy to localize computational work for the efficient use with future extreme-s...

A Newton-Krylov-FETI-DP method for solving problems in elastoplasticity is considered. In some cases additional coarse constraints are necessary to guarantee good convergence of the pcg algorithm. To enhance the coarse space in the FETI-DP method, we use a strategy introduced in Mandel and Sousedík (Comput. Methods Appl. Mech. Eng. 196, 1389–1399,...

We describe a BDDC algorithm, see e.g., [1], and an adaptive coarse space enforced by a transformation of basis for the iterative solution of scalar diffusion problems with a discontinuous diffusion coefficient. The coefficient varies over several orders of magnitude both inside of the subdomains and along the interface. A related algorithm for FET...

The accurate prediction of transmural stresses in arterial walls requires on the one hand robust and efficient numerical schemes for the solution of boundary value problems including fluid-structure interactions (FSI) and on the other hand the use of a material model for the vessel wall which is able to capture the relevant features of the material...

A Fluid–Structure Interaction (FSI) problem can be reinterpreted as a heterogeneous problem with two subdomains. It is possible to describe the coupled problem at the interface between the fluid and the structure, yielding a nonlinear Steklov–Poincaré problem. The linear system can be linearized by Newton iterations on the interface and the resulti...

A special finite element method based on approximate component mode synthesis (ACMS) was introduced in Hetmaniuk and Lehoucq (2010). ACMS was developed for second order elliptic partial differential equations with rough or highly varying coefficients. Here, a parallel implementation of ACMS is presented and parallel scalability issues are discussed...

A coarse space is constructed for the dual-primal finite element tearing and interconnecting (FETI-DP) domain decomposition method applied to highly heterogeneous problems by solving local generalized eigenvalue problems. For certain problems with highly varying coefficients, e.g., from multiscale simulations, the coefficient jump will appear in th...

A new coarse space for FETI-DP domain decomposition methods for mixed finite element discretizations of almost incompressible linear elasticity problems in 3D is presented. The mixed finite element discretization uses continuous piecewise triquadratic displacements and discontinuous piecewise constant pressures. The piecewise constant pressure vari...

The solution of nonlinear problems, e.g., in material science, requires fast and highly scalable parallel solvers. Finite element tearing and interconnecting dual primal (FETI-DP) domain decomposition methods are parallel solution methods for implicit problems discretized by finite elements. Recently, nonlinear versions of the well-known FETI-DP me...

We present an approach to hybrid MPI/OpenMP parallelization in FETI-DP methods using OpenMP with PETSc+MPI in the finite element assembly and using the shared memory parallel direct solver Pardiso in the FETI-DP solution phase. Our approach thus uses OpenMP parallelization on subdomains and MPI in between subdomains. We investigate the efficiency o...

An adaptive coarse space for the FETI-DP domain decomposition method based on generalized eigenvalue problems is presented. For details on the FETI-DP algorithm, see, e.g., [1]. The method is a variation of a method proposed in [2, 3], where deluxe scaling was used. Here, an extension is presented, which allows several other scalings. (© 2014 Wiley...

A parallel FETI-DP (Finite Element Tearing and Interconnecting) domain decomposition method is applied to optimal control problems with control constraints. We show parallel scalability for up to 1024 cores of a Cray XT6. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

Two different aspects of FETI-DP domain decomposition methods are considered. In the first part, the adaptive construction of coarse spaces from local eigenvalue problems for the solution of heterogeneous, e.g., multiscale, problems is considered. This strategy to construct a coarse space is implemented using a deflation approach. In the second par...

FETI-DP methods for the optimal control problems of linear elasticity problems are considered and numerical results are presented.

New nonlinear FETI-DP (dual-primal finite element tearing and interconnecting) and BDDC (balancing domain decomposition by constraints) domain decomposition methods are introduced. In all these methods, in each iteration, local nonlinear problems are solved on the subdomains. The new approaches can significantly reduce communication and show a sign...

The simulation of the physiological loading situation of arteries with moderate atherosclerotic plaque may provide additional indicators for medical doctors to estimate if the plaque is likely to rupture and if surgical intervention is required. In particular the transmural stresses are important in this context. They depend strongly on the mechani...

The purpose of this article is to present convergence bounds and some preliminary numerical results for a special category of problems of compressible and almost incompressible linear elasticity when using FETI-DP or BDDC domain decomposition methods.

In this paper, we consider the elastic deformation of arterial walls as occurring, e.g., in the process of a balloon angioplasty, a common treatment in the case of atherosclerosis. Soft biological tissue is an almost incompressible material. To account for this property in finite element simulations commonly used free energy functions contain terms...

Arterial walls in the healthy physiological regime are characterized by quasi-incompressible, anisotropic, hyperelastic material behavior. Polyconvex material functions representing such materials typically incorporate a penalty function to account for the incompressibility. Unfortunately, the penalty will affect the conditioning of the stiffness m...

Numerical results of nonlinear parallel simulations of soft biological tissue are shown using the FETI-DP method. An hyperelastic model is considered which incorporates almost incompressibility as well as anisotropy. An arterial geometry reconstructed from ultrasound data is used. Numerical scalability is shown using up to 4096 cores of a Cray XT6....

Purpose The purpose of this paper is to present a computational framework for the simulation of patient‐specific atherosclerotic arterial walls. Such simulations provide information regarding the mechanical stress distribution inside the arterial wall and may therefore enable improved medical indications for or against medical treatment. In detail,...

Two strategies, using edge averages, for FETI-DP (dual–primal finite element tearing and interconnecting) methods for contact problems are considered. The first one is a preconditioning technique by a conjugate projector, where the Lagrange multipliers corresponding to the variables of the coinciding edges are aggregated. The second one is an expli...

FETI-DP (dual-primal finite element tearing and interconnecting) methods are nonoverlapping domain decomposition methods which are used to solve large algebraic systems of equations that arise, e.g., from problems in linear elasticity. Good convergence bounds for problems of compressible linear elasticity are well known for two-and three-dimensiona...

In this paper, projector preconditioning, also known as the deflation method, as well as the balancing preconditioner are applied to the dual-primal finite element tearing and interconnecting (FETI-DP) and balancing domain decomposition by constraints (BDDC) methods in order to create a second, independent coarse problem. This may help to extend th...

We consider linear elliptic systems which arise in coupled elastic continuum mechanical models. In these systems, the strain tensor ε P := sym (P -1∇u) is redefined to include a matrix valued inhomogeneity P(x) which cannot be described by a space dependent fourth order elasticity tensor. Such systems arise naturally in geometrically exact plastic...

Biological soft tissues appearing in arterial walls are characterized by a nearly incompressible, anisotropic, hyperelastic material behavior in the physiological range of deformations. For the representation of such materials we apply a polyconvex strain energy function in order to ensure the existence of minimizers and in order to satisfy the Leg...

A minimization problem modeling geometrically exact generalized continua of micromorphic type is considered. The solution consists of two fields, the elastic deformation φ of a given body and a tensorial field P which can model different additional features needed for a more reliable description of solids. For the solution of this minimization prob...

Highly scalable parallel domain decomposition methods for elliptic partial differential equations are considered with a special emphasis on problems arising in elasticity. The focus of this survey article is on Finite Element Tearing and Interconnecting (FETI) methods, a family of nonoverlapping domain decomposition methods where the continuity bet...