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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-dimensional problems. More recently, FETI-DP and BDDC (balancing domain decomposition by constraints) methods have been developed that are robust also in the regime of homogeneous almost incompressible linear elasticity. The coarse space of such methods is large especially in 3D (three dimensions) and its implementation needs knowledge of geometrical information. Here, the convergence of FETI-DP methods for problems in 3D with almost incompressible inclusions or compressible inclusions with different material parameters embedded in a compressible matrix material is analyzed. For such problems, where the material is compressible in the vicinity of the subdomain interface, a polylogarithmic condition number estimate is shown for the preconditioned FETI-DP system. This bound depends only on the thickness of the compressible hull but is otherwise independent of coefficient jumps between subdomains and also between the hull and the inclusion. The bound is also valid for corresponding BDDC methods. The new contribution of the current paper is a theory that provides condition number bounds for the case of varying incompressibility and also varying Young moduli inside subdomains without changing the coarse space. [ABSTRACT FROM AUTHOR]

We consider linear elliptic systems which arise in coupled elastic continuum mechanical models. In these systems, the strain tensor εP:= 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 plasticity or in problems with eigenstresses. The tensor field Pinduces a structural change of the elasticity equations. For such a model the FETI-DP method is formulated and a convergence estimate is provided for the special case that P-T= ∇ψis a gradient. It is shown that the condition number depends only quadratic-logarithmically on the number of unknowns of each subdomain. The dependence of the constants of the bound on Pis highlighted. Numerical examples confirm our theoretical findings. Promising results are also obtained for settings which are not covered by our theoretical estimates. [ABSTRACT FROM AUTHOR]

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 between the subdomains, in principle, is enforced by the use of Lagrange multipliers. Exact onelevel and dual-primal FETI methods as well as related inexact dual-primal variants are described and theoretical convergence estimates are presented together with numerical results confirming the parallel scalability properties of these methods. New aspects such as a hybrid onelevel FETI/FETI-DP approach and the behavior of FETI-DP for anisotropic elasticity problems are presented. Parallel and numerical scalability of the methods for more than 65 000 processor cores of the JUGENE supercomputer is shown. An application of a dual-primal FETI method to a nontrivial biomechanical problem from nonlinear elasticity, modeling arterial wall stress, is given, showing the robustness of our domain decomposition methods for such problems. [ABSTRACT FROM AUTHOR]

In the theory for domain decomposition algorithms of the iterative substructuring family, each subdomain is typically assumed to be the union of a few coarse triangles or tetrahedra. This is an unrealistic assumption, in particular if the subdomains result from the use of a mesh partitioner, in which case they might not even have uniformly Lipschitz continuous boundaries. The purpose of this study is to derive bounds for the condition number of these preconditioned conjugate gradient methods which depend only on a parameter in an isoperimetric inequality, two geometric parameters characterizing John and uniform domains, and the maximum number of edges of any subdomain. A related purpose is to explore to what extent well-known technical tools previously developed for quite regular subdomains can be extended to much more irregular subdomains. Some of these results are valid for any John domain, while an extension theorem, which is needed in this study, requires that the subdomains have complements which are uniform. The results, so far, are complete only for problems in two dimensions. Details are worked out for a FETI-DP algorithm and numerical results support the findings. Some of the numerical experiments illustrate that care must be taken when selecting the scaling of the preconditioners in the case of irregular subdomains. [ABSTRACT FROM AUTHOR]

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 to obtain symmetric matrices, which are typically better suited for iterative solvers. The MPI-parallel implementation uses the software libraries deal.II, p4est and FROSch (Fast of Robust Overlapping Schwarz). FROSch is part of the Trilinos library and is used in fully algebraic mode, i.e., the preconditioner is constructed from the monolithic system matrix without making explicit use of the problem structure. Strong and weak parallel scalability is studied using up to 512 cores, considering the standard GDSW (Generalized Dryja-Smith-Widlund) coarse space and the newer coarse space with reduced dimension. The FROSch solver is applicable to the coupled problems within in the range of processor cores considered here, although numerical scalablity cannot be expected (and is not observed) for the fully algebraic mode. In our strong scalability study, the average number of Krylov iterations per Newton iteration is higher by a factor of up to six compared to a linear elasticity problem. However, making mild use of the problem structure in the preconditioner, this number can be reduced to a factor of two and, importantly, also numerical scalability can then be achieved experimentally. Nevertheless, the fully algebraic mode is still preferable since a faster time to solution is achieved. [ABSTRACT FROM AUTHOR]

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 prediction of target events. However, they typically provide little insight into the problem regarding the importance, or relevance, of covariables. On the other hand, logistic regression has a well understood statistical foundation and uses an explicit model from which knowledge can be gained about the underlying phenomenon. Establishing such an explicit model is rather difficult for real world problems. In the context of mineral prospectivity modeling additional challenges arise, such as rare events, i.e. only a small fraction of data instances describes a positive target event, which is the event of interest. In this paper, we propose a model selection procedure applied to logistic regression incorporating explicit nonlinearities. The model selection procedure, based on the Wald test and the Bayes' information criterion (BIC), as proposed in this paper is new. The performance regarding the predictive power of the obtained model is comparable to logistic regression using a stepwise model selection and to neural networks on several real world datasets, one of them a dataset for the detection of gold mineralizations in Ghana. However, our new method is significantly faster than standard stepwise selection, while selecting fewer variables for the final model. In our numerical experiments, the prediction accuracy is also comparable to a neural network, which is currently in use in industry. In applications, the method can aid the model building process through an explicit model. Furthermore, it may be used as preprocessing step for other machine learning algorithms such as neural networks. In this paper, we intend to present mathematics of prospectivity modeling with the potential to contribute to bridging the gap between statistical and machine learning. Big Data and Artificial Intelligence are of increasing importance in mineral exploration. At the same time there is a growing demand for mathematically rigorous machine learning methods, which can still be interpreted by experts. This paper is a contribution to this field. [ABSTRACT FROM AUTHOR]

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 presented. [ABSTRACT FROM AUTHOR]

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 , can be used for the simulation of the Nakajima test. In this test, a sample sheet geometry is clamped between a blank holder and a die. Then, a hemispherical punch is driven into the specimen until material failure occurs. For the simulation of the Nakajima test, our software package FE2TI has been enhanced with a frictionless contact formulation on the macroscopic level using the penalty method. The appropriate choice of suitable boundary conditions as well as the influence of symmetry assumptions regarding the symmetric test setup are discussed. In order to be able to solve larger macroscopic problems more efficiently, the balancing domain decomposition by constraints (BDDC) approach has been implemented on the macroscopic level as an alternative to a sparse direct solver. To improve the computational efficiency of FE2TI even further, additionally, an adaptive load step approach has been implemented and different extrapolation strategies are compared. Both strategies yield a significant reduction of the overall computing time. Furthermore, a strategy to dynamically increase the penalty parameter is presented which allows to resolve the contact conditions more accurately without increasing the overall computing time too much. Numerically computed forming limit diagrams based on virtual Nakajima tests are presented. [ABSTRACT FROM AUTHOR]

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 muscle activation. The coupled deformation‐diffusion problem is then solved using the finite element software FEDDLib and overlapping Schwarz preconditioners from the Trilinos package FROSch. These preconditioners include highly scalable parallel GDSW (generalized Dryja–Smith–Widlund) and RGDSW (reduced GDSW) preconditioners. Simulation results show the expected increase in the lumen diameter of an idealized artery due to the drug‐induced reduction of smooth muscle contraction, as well as a decrease in the rate of arterial contraction in the presence of calcium channel blockers. Strong and weak parallel scalability of the resulting computational implementation are also analyzed. [ABSTRACT FROM AUTHOR]

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 simulated for ensembles of microstructure samples. We determine effective material parameters, i.e., the effective Lamé moduli λ ∗ and μ ∗ , on the macroscale by fitting a macroscopic material model to the microscopic stress data, using stress averaging over many microstructure samples. The effective parameters λ ∗ and μ ∗ are considered as functions of the microscale material parameters and the geometric parameters of the Boolean model including the grain shape. We also consider the size of the Representative Volume Element (RVE) given a precision and an ensemble size. We use structured and unstructured meshes and also provide a comparison with the FE 2 method. [ABSTRACT FROM AUTHOR]

Parallel computational homogenization using the well-knwon 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 FE 2 method can take advantage of the largest supercomputers available and those of the upcoming exascale era for virtual material testing of micro-heterogeneous materials such as advanced steel. The FE 2 method is a computational micro-macro homogenization approach where at each Gauss integration point of the macroscopic finite element problem a microscopic finite element problem, defined on a representative volume element (RVE), is attached. Note that the FE 2 method is not embarrassingly parallel since the RVE problems are coupled through the macroscopic problem. Numerical results considering different grids on both, the macroscopic and microscopic level as well as weak scaling results for up to a million parallel processes are presented. [ABSTRACT FROM AUTHOR]