Adaptive BDDC in Three Dimensions

The adaptive BDDC method is extended to the selection of face constraints in three dimensions. A new implementation of the BDDC method is presented based on a global formulation without an explicit coarse problem, with massive parallelism provided by a multifrontal solver. Constraints are implemented by a projection and sparsity of the projected operator is preserved by a generalized change of variables. The effectiveness of the method is illustrated on several engineering problems.Comment: 28 pages, 9 figures, 9 table

Improved recovery of admissible stress in domain decomposition methods - application to heterogeneous structures and new error bounds for FETI-DP

This paper investigates the question of the building of admissible stress field in a substructured context. More precisely we analyze the special role played by multiple points. This study leads to (1) an improved recovery of the stress field, (2) an opportunity to minimize the estimator in the case of heterogeneous structures (in the parallel and sequential case), (3) a procedure to build admissible fields for FETI-DP and BDDC methods leading to an error bound which separates the contributions of the solver and of the discretization

Combining Machine Learning and Adaptive Coarse Spaces - A Hybrid Approach for Robust FETI-DP Methods in Three Dimensions

The hybrid ML-FETI-DP algorithm combines the advantages of adaptive coarse spaces in domain decomposition methods and certain supervised machine learning techniques. Adaptive coarse spaces ensure robustness of highly scalable domain decomposition solvers, even for highly heterogeneous coefficient distributions with arbitrary coefficient jumps. However, their construction requires the setup and solution of local generalized eigenvalue problems, which is typically computationally expensive. The idea of ML-FETI-DP is to interpret the coefficient distribution as image data and predict whether an eigenvalue problem has to be solved or can be neglected while still maintaining robustness of the adaptive FETI-DP method. For this purpose, neural networks are used as image classifiers. In the present work, the ML-FETI-DP algorithm is extended to three dimensions, which requires both a complex data preprocessing procedure to construct consistent input data for the neural network as well as a representative training and validation data set to ensure generalization properties of the machine learning model. Numerical experiments for stationary diffusion and linear elasticity problems with realistic coefficient distributions show that a large number of eigenvalue problems can be saved; in the best case of the numerical results presented here, 97% of the eigenvalue problems can be avoided to be set up and solved

Substructured formulations of nonlinear structure problems - influence of the interface condition

We investigate the use of non-overlapping domain decomposition (DD) methods for nonlinear structure problems. The classic techniques would combine a global Newton solver with a linear DD solver for the tangent systems. We propose a framework where we can swap Newton and DD, so that we solve independent nonlinear problems for each substructure and linear condensed interface problems. The objective is to decrease the number of communications between subdomains and to improve parallelism. Depending on the interface condition, we derive several formulations which are not equivalent, contrarily to the linear case. Primal, dual and mixed variants are described and assessed on a simple plasticity problem.Comment: in International Journal for Numerical Methods in Engineering, Wiley, 201

Simultaneous-FETI and Block-FETI: robust domain decomposition with multiple search directions.

International audienceDomain Decomposition methods often exhibit very poor performance when applied to engineering problems with large heterogeneities. In particular for heterogeneities along domain interfaces the iterative techniques to solve the interface problem are lacking an efficient preconditioner. Recently a robust approach, named FETI-Geneo, was proposed where troublesome modes are precomputed and deflated from the interface problem. The cost of the FETI-Geneo is however high. We propose in this paper techniques that share similar ideas with FETI-Geneo but where no pre-processing is needed and that can be easily and efficiently implemented as an alternative to standard Domain Decomposition methods. In the block iterative approaches presented in this paper, the search space at every iteration on the interface problem contains as many directions as there are domains in the decomposition. Those search directions originate either from the domain-wise preconditioner (in the Simultaneous FETI method) or from the block structure of the right-hand side of the interface problem (Block FETI). We show on 2D structural examples that both methods are robust and provide good convergence in the presence of high heterogeneities, even when the interface is jagged or when the domains have a bad aspect ratio. The Simultaneous FETI was also efficiently implemented in an optimized parallel code and exhibited excellent performance compared to the regular FETI method

Learning Adaptive FETI-DP Constraints for Irregular Domain Decompositions

Adaptive coarse spaces yield a robust convergence behavior for FETI-DP (Finite Element Tearing and Interconnecting - Dual Primal) and BDDC (Balancing Domain Decomposition by Constraints) methods for highly heterogeneous problems. However, the usage of such adaptive coarse spaces can be computationally expensive since, in general, it requires the setup and the solution of a relatively high amount of local eigenvalue problems on parts of the domain decomposition interface. In earlier works, see, e.g., [2], it has been shown that it is possible to train a neural network to make an automatic decision which of the eigenvalue problems in an adaptive FETI-DP method are actually necessary for robustness with a satisfactory accuracy. Moreover, these results have been extended in [6] by directly learning an approximation of the adaptive edge constraints themselves for regular, two-dimensional domain decompositions. In particular, this does not require the setup or the solution of any eigenvalue problems at all since the FETI-DP coarse space is, in this case, exclusively enhanced by the learned constraints obtained from the regression neural networks trained in an offline phase. Here, in contrast to [6], a regression neural network is trained with both, training data resulting from straight and irregular edges. Thus, it is possible to use the trained networks also for the approximation of adaptive constraints for irregular domain decompositions. Numerical results for a heterogeneous two-dimensional stationary diffusion problem are presented using both, a decomposition into regular and irregular subdomains

Balancing domain decomposition by constraints and perturbation

In this paper, we formulate and analyze a perturbed formulation of the balancing domain decomposition by constraints (BDDC) method. We prove that the perturbed BDDC has the same polylogarithmic bound for the condition number as the standard formulation. Two types of properly scaled zero-order perturbations are considered: one uses a mass matrix, and the other uses a Robin-type boundary condition, i.e, a mass matrix on the interface. With perturbation, the wellposedness of the local Neumann problems and the global coarse problem is automatically guaranteed, and coarse degrees of freedom can be defined only for convergence purposes but not well-posedness. This allows a much simpler implementation as no complicated corner selection algorithm is needed. Minimal coarse spaces using only face or edge constraints can also be considered. They are very useful in extreme scale calculations where the coarse problem is usually the bottleneck that can jeopardize scalability. The perturbation also adds extra robustness as the perturbed formulation works even when the constraints fail to eliminate a small number of subdomain rigid body modes from the standard BDDC space. This is extremely important when solving problems on unstructured meshes partitioned by automatic graph partitioners since arbitrary disconnected subdomains are possible. Numerical results are provided to support the theoretical findings.Peer ReviewedPostprint (published version

Combining Machine Learning and Domain Decomposition Methods – A Review

Scientific machine learning, an area of research where techniques from machine learning and scientific computing are combined, has become of increasing importance and receives growing attention. Here, our focus is on a very specific area within scientific machine learning given by the combination of domain decomposition methods with machine learning techniques. The aim of the present work is to make an attempt of providing a review of existing and also new approaches within this field as well as to present some known results in a unified framework; no claim of completeness is made. As a concrete example of machine learning enhanced domain decomposition methods, an approach is presented which uses neural networks to reduce the computational effort in adaptive domain decomposition methods while retaining their robustness. More precisely, deep neural networks are used to predict the geometric location of constraints which are needed to define a robust coarse space. Additionally, two recently published deep domain decomposition approaches are presented in a unified framework. Both approaches use physics-constrained neural networks to replace the discretization and solution of the subdomain problems of a given decomposition of the computational domain. Finally, a brief overview is given of several further approaches which combine machine learning with ideas from domain decomposition methods to either increase the performance of already existing algorithms or to create completely new methods