Balancing Domain Decomposition Research Articles

A comparative study of scalable multilevel preconditioners for cardiac mechanics

In this work, we provide a performance comparison between the Balancing Domain Decomposition by Constraints (BDDC) and the Algebraic Multigrid (AMG) preconditioners for cardiac mechanics on both structured and unstructured finite element meshes. The mechanical behavior of myocardium can be described by the equations of three-dimensional finite elasticity, which are discretized by finite elements in space and yield the solution of a large scale nonlinear algebraic system. This problem is solved by a Newton-Krylov method, where the solution of the Jacobian linear system is accelerated by BDDC/AMG preconditioners. We thoroughly explore the main parameters of the BDDC preconditioner in order to make the comparison fair. We focus on: the performance of different direct solvers for the local and coarse problems of the BDDC algorithm; the impact of the different choices of BDDC primal degrees of freedom; and the influence of the finite element degree. Scalability tests are performed on Linux clusters up to 1024 processors, and we conclude with a performance study on a realistic electromechanical simulation.

Balancing domain decomposition method for large-scale analysis of an assembly structure having millions of multipoint constraints

A method by which to incorporate multipoint constraints (MPCs) into the balancing domain decomposition (BDD) method proposed in a previous study (BDD-MPC method) is investigated, improved, and applied to realistic and complex problems. First, the effects of the penalty term to take into account the MPCs in the coarse-grid problem of the BDD method are investigated. Then, improved implementation techniques for the BDD-MPC method are proposed. The computation speed of the projection procedure in the conjugate projected gradient method is improved using the block diagonal structure of the constraint matrix. The parallel solution procedure for the coarse-grid problem considering the MPCs that are projected to the coarse space is also improved. The improved MPC-BDD method is implemented in ADVENTURE_Solid, which is a parallel finite element structural analysis code. The code is parallelized using an MPI-OpenMP hybrid parallel implementation framework. In the previous study, the BDD-MPC method was applied to only simple numerical examples. In the present study the performance of the improved method is demonstrated by solving super-high-rise building models having very complex geometry and several million MPCs.

BDDC Algorithms for Oseen problems with HDG Discretizations

Abstract The balancing domain decomposition by constraints (BDDC) methods are applied to the linear system arising from the hybridizable discontinuous Galerkin (HDG) discretization of the Oseen equation of incompressible flow. The generalized minimal residual method (GMRES) is used to accelerate the convergence. The original system is first reduced to a subdomain interface problem that is asymmetric indefinite, but can be positive definite in a special subspace. Edge/face average constraints can ensure all BDDC-preconditioned GMRES iterates stay in this special subspace. The convergence of the algorithm is analyzed, and additional edge/face constraints are used to improve the convergence. If the subdomain size is small enough, the number of iterations is independent of the number of subdomains, and depends only slightly on the subdomain problem size when the viscosity is large. When the viscosity is small, the convergence deteriorates. However, the numerical examples can give satisfactory results for the linear HDG discretization even with a small viscosity.

Newton–Krylov-BDDC deluxe solvers for non-symmetric fully implicit time discretizations of the bidomain model

A novel theoretical convergence rate estimate for a Balancing Domain Decomposition by Constraints algorithm is proven for the solution of the cardiac bidomain model, describing the propagation of the electric impulse in the cardiac tissue. The non-linear system arises from a fully implicit time discretization and a monolithic solution approach. The preconditioned non-symmetric operator is constructed from the linearized system arising within the Newton–Krylov approach for the solution of the non-linear problem; we theoretically analyze and prove a convergence rate bound for the Generalised Minimal Residual iterations’ residual. The theory is confirmed by extensive parallel numerical tests, widening the class of robust and efficient solvers for implicit time discretizations of the bidomain model.

Acceleration of a parallel BDDC solver by using graphics processing units on subdomains

An approach to accelerating a parallel domain decomposition (DD) solver by graphics processing units (GPUs) is investigated. The solver is based on the Balancing Domain Decomposition Method by Constraints (BDDC), which is a nonoverlapping DD technique. Two kinds of local matrices are required by BDDC. First, dense matrices corresponding to local Schur complements of interior unknowns are constructed by the sparse direct solver. These are further used as part of the local saddle-point problems within BDDC. In the next step, the local matrices are copied to GPUs. Repeated multiplications of local vectors with the dense matrix of the Schur complement are performed for each subdomain. In addition, factorizations and backsubstitutions with the dense saddle-point subdomain matrices are also performed on GPUs. Detailed times of main components of the algorithm are measured on a benchmark Poisson problem. The method is also applied to an unsteady problem of incompressible flow, where the Krylov subspace iterations are performed repeatedly in each time step. The results demonstrate the potential of the approach to speed up realistic simulations up to 5 times with a preference towards large subdomains.

Robust BDDC algorithms for the Brinkman problem with HDG discretizations

The balancing domain decomposition by constraints methods (BDDC) have been applied to solve the saddle point problem arising from a hybridizable discontinuous Galerkin (HDG) discretization for the Brinkman equations. Edge/face average constraints are enforced across the subdomain interface for each velocity component to ensure that the BDDC preconditioned conjugate gradient (CG) iterations stay in a special subspace, where the reduced system from the original saddle point problem is positive definite. The condition number is proved to be uniformly bounded under both Stokes and Darcy dominant cases with a deluxe scaling for the parameters with jumps only across the subdomain interface. When the parameters are highly discontinuous with large jumps inside each subdomains, additional adaptively chosen primal constraints, obtained by solving local generalized eigenvalue problems, are introduced to control the condition numbers. Numerical experiments confirm the theory.

BDDC Deluxe Algorithms for Two-Dimensional H($curl$) Isogeometric Analysis

Isogeometric analysis has been introduced as an alternative to finite elements to simplify the integration of computer-aided design and the discretization of variational problems of continuum mechanics. In contrast to finite elements, the basis functions of isogeometric analysis, B-splines and nonuniform rational basis splines, are typically not nodal which makes the interface between subdomains fat with several layers of knots and which creates new issues in the design and analysis of iterative solvers based on domain decomposition methods. The resulting systems of algebraic equations also tend to be much more ill-conditioned than those derived from finite elements. The deluxe variant of adaptive balancing domain decomposition by constraints algorithms have proven successful, and in this study, it is applied to two-dimensional problems formulated in H($curl$). Numerical results show very good performance even for high degrees of the underlying B-splines, and fast convergence has been found for $p \leq 7$ using no more than $2(p^2+p-1)$ global (primal) variables per subdomain, where $p$ is the degree of the B-splines. All our results are focused on the most difficult case with the maximum degree of continuity of the B-splines.

BDDC Preconditioners for Divergence Free Virtual Element Discretizations of the Stokes Equations

The virtual element method (VEM) is a new family of numerical methods for the approximation of partial differential equations, where the geometry of the polytopal mesh elements can be very general. The aim of this article is to extend the balancing domain decomposition by constraints preconditioner to the solution of the saddle-point linear system arising from a VEM discretization of the two-dimensional Stokes equations. Under suitable hypotesis on the choice of the primal unknowns, the preconditioned linear system results symmetric and positive definite, thus the preconditioned conjugate gradient method can be used for its solution. We provide a theoretical convergence analysis estimating the condition number of the preconditioned linear system. Several numerical experiments validate the theoretical estimates, showing the scalability and quasi-optimality of the method proposed. Moreover, the solver exhibits a robust behavior with respect to the shape of the polygonal mesh elements. We also show that a faster convergence could be achieved with an easy to implement coarse space, slightly larger than the minimal one covered by the theory.

Implementation of balancing domain decomposition method for parallel finite element analysis involving inactive elements

AbstractThe present study developed a numerical method to implement domain decomposition (DD) solvers with the diagonal‐scaling preconditioner, the balancing domain decomposition (BDD) preconditioner and the BDD with diagonal scaling (BDD‐DIAG) preconditioner, for inactive elements. The inactive element, which is a finite element having zero stiffness, is used in several fields such as multi‐pass welding analysis, additive manufacturing analysis, damage analysis, and topology optimization. For this sort of analysis, we adopted the one‐time decomposition approach, in which the DD process is performed once at the beginning of the analysis. Based on this approach, we formulated the matrix–vector multiplication, the preconditioning and the vector operations in the algorithm of the conjugate gradient method, along with the inactive elements and floating degrees of freedom caused by the inactive elements. Consideration of the inactive elements is enabled by the slight modifications of matrices and vectors in the algorithm. Numerical examples confirmed the scalability of the BDD and BDD‐DIAG preconditioners with the present implementation method. Moreover, the capability of the present method for damage analysis, topology computation, and thermal elastic–plastic analysis of metal additive manufacturing problems was demonstrated.

Parallel Newton--Krylov BDDC and FETI-DP Deluxe Solvers for Implicit Time discretizations of the Cardiac Bidomain Equations

Two novel parallel Newton-Krylov Balancing Domain Decomposition by Constraints (BDDC) and Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) solvers are here constructed, analyzed and tested numerically for implicit time discretizations of the three-dimensional Bidomain system of equations. This model represents the most advanced mathematical description of the cardiac bioelectrical activity and it consists of a degenerate system of two non-linear reaction-diffusion partial differential equations (PDEs), coupled with a stiff system of ordinary differential equations (ODEs). A finite element discretization in space and a segregated implicit discretization in time, based on decoupling the PDEs from the ODEs, yields at each time step the solution of a non-linear algebraic system. The Jacobian linear system at each Newton iteration is solved by a Krylov method, accelerated by BDDC or FETI-DP preconditioners, both augmented with the recently introduced {\em deluxe} scaling of the dual variables. A polylogarithmic convergence rate bound is proven for the resulting parallel Bidomain solvers. Extensive numerical experiments on linux clusters up to two thousands processors confirm the theoretical estimates, showing that the proposed parallel solvers are scalable and quasi-optimal.

Robust and scalable adaptive BDDC preconditioners for virtual element discretizations of elliptic partial differential equations in mixed form

The Virtual Element Method (VEM) is a recent numerical technology for the solution of partial differential equations on computational grids constituted by polygonal or polyhedral elements of very general shape. The aim of this work is to develop effective linear solvers for a general order VEM approximation designed to approximate three-dimensional scalar elliptic equations in mixed form. The proposed Balancing Domain Decomposition by Constraints (BDDC) preconditioner allows to use conjugate gradient iterations, albeit the algebraic linear systems arising from the discretization of the problem are indefinite, ill-conditioned, and of saddle point nature. The condition number of the resulting positive definite preconditioned system is adaptively controlled by means of deluxe scaling operators and suitable local generalized eigenvalue problems for the selection of optimal primal constraints. Numerical results confirm the theoretical estimates and the reliability of the adaptive procedure, with the experimental condition numbers always very close to the prescribed adaptive tolerance parameter. The scalability and quasi-optimality of the preconditioner are demonstrated, and the performances of the proposed solver are compared with state-of-the-art parallel direct solvers and block preconditioning techniques in a distributed memory setting.

離散deRham系列を満たす高次多角形要素を用いた静磁場問題に対する領域分割法

A Balancing Domain Decomposition (BDD) method is considered as the preconditioner of an iterative Domain Decomposition Method (DDM) for perturbed magnetostatic problems, where the magnetic vector potential is regarded as an unknown function approximated by the Ned´ elec curl conforming finite element. To reduce the number of the Degrees´ Of Freedom (DOF) of coarse spaces in BDD methods, Polynomial Element Methods (PEMs) are introduced. Owing to the introduction of PEMs and the result of BDD method originally proposed by Mandel, the condition number of coefficient matrices derived from the iterative DDM is evaluated. Therefore, the number of iterations of the iterative DDM can be kept even when the number of the subdomains becomes larger. Moreover, the approximate coarse space by PEMs can admit that each subdomain becomes more general polyhedron. The results in case of higher order PEMs are also shown.

Robust BDDC algorithms for finite volume element methods

The balancing domain decomposition by constraints (BDDC) method is applied to the linear system arising from the finite volume element method (FVEM) discretization of a scalar elliptic equation. The FVEMs share nice features of both finite element and finite volume methods and are flexible for complicated geometries with good conservation properties. However, the resulting linear system usually is asymmetric. The generalized minimal residual (GMRES) method is used to accelerate convergence. The proposed BDDC methods allow for jumps of the coefficient across subdomain interfaces. When jumps of the coefficient appear inside subdomains, the BDDC algorithms adaptively choose the primal variables deriving from the eigenvectors of some local generalized eigenvalue problems. The adaptive BDDC algorithms with advanced deluxe scaling can ensure good performance with highly discontinuous coefficients. A convergence analysis of the BDDC method with a preconditioned GMRES iteration is provided, and several numerical experiments confirm the theoretical estimate.

領域分割法と簡略化熱源モデルを用いた金属積層造形問題の熱弾塑性解析とバリデーション

The metal additive manufacturing problems can be modeled by the one way coupling problem from a heat conduction to a thermal elastic plastic analysis. It costs a lot of time and temperature steps to analyze the problems. To reduce the numbers of these steps, our previous studies used the balancing domain decomposition method as a parallel strategy, and a simplified heat source model. The model allows to sequentially heat an entire layer one by one, and this process continues until all layers are simulated. However, it was only applied to a simulation of depositing a plate. Thus, this study focuses on applying the model to a shape other than a plate with the balancing domain decomposition method. Especially, a half pipe model was considered. The paper presents how to find parameters of the simplified heat source model on the half pipe model. One of numerical results showed a speedup of more than one hundred times. They were also verified and validated.

BDDC for MHFEM discretization of unsteady two-phase flow in porous media

This work deals with the application of the Balancing Domain Decomposition by Constrains (BDDC) method to unsteady two-phase flow problems in porous media. We briefly describe the spatial discretization of the problem which is based on the mixed-hybrid finite element method (MHFEM) and semi-implicit time discretization. Then, we describe the BDDC method, in detail discuss the differences between the symmetric and nonsymmetric cases, and present the necessary modifications of the algorithm for the more complicated nonsymmetric case. We describe the parallel implementation of the method and highlight the critical steps of the algorithm that affect the performance and scalability. The parallel implementation is then tested on benchmark problems in 2D and 3D and its efficiency is investigated on various meshes. The numerical results indicate that the method preserves good computational efficiency for increasing number of processes and, therefore, allows solving problems on very fine meshes. In the case of unsteady problem, additional speed-up is achieved using the information from previous time steps for the solution in the current time step.

Robust Discretization and Solvers for Elliptic Optimal Control Problems with Energy Regularization

Abstract We consider elliptic distributed optimal control problems with energy regularization. Here the standard L 2 {L_{2}} -norm regularization is replaced by the H - 1 {H^{-1}} -norm leading to more focused controls. In this case, the optimality system can be reduced to a single singularly perturbed diffusion-reaction equation known as differential filter in turbulence theory. We investigate the error between the finite element approximation u ϱ ⁢ h {u_{\varrho h}} to the state u and the desired state u ¯ {\overline{u}} in terms of the mesh-size h and the regularization parameter ϱ. The choice ϱ = h 2 {\varrho=h^{2}} ensures optimal convergence the rate of which only depends on the regularity of the target function u ¯ {\overline{u}} . The resulting symmetric and positive definite system of finite element equations is solved by the conjugate gradient (CG) method preconditioned by algebraic multigrid (AMG) or balancing domain decomposition by constraints (BDDC). We numerically study robustness and efficiency of the AMG preconditioner with respect to h, ϱ, and the number of subdomains (cores) p. Furthermore, we investigate the parallel performance of the BDDC preconditioned CG solver.

BDDC algorithms for advection-diffusion problems with HDG discretizations

In this paper, a preconditioned GMRES method is developed and analyzed for solving the linear system from advection-diffusion equations with the hybridizable discontinuous Galerkin (HDG) discretization. The preconditioner is the balancing domain decomposition methods (BDDC), one of the most popular nonoverlapping domain decomposition methods. For large viscosity, if the subdomain size is small enough, the number of iterations is independent of the number of subdomains and depends only slightly on the subdomain problem size. The convergence deteriorates when the viscosity decreases. These results are similar to those with the standard finite element discretizations. Numerical results of two examples in two dimensions are provided to confirm the theory.

Fully-coupled micro\u2013macro finite element simulations of the Nakajima test using parallel computational homogenization

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.

Learning Adaptive Coarse Spaces of BDDC Algorithms for Stochastic Elliptic Problems with Oscillatory and High Contrast Coefficients

In this paper, we consider the balancing domain decomposition by constraints (BDDC) algorithm with adaptive coarse spaces for a class of stochastic elliptic problems. The key ingredient in the construction of the coarse space is the solutions of local spectral problems, which depend on the coefficient of the PDE. This poses a significant challenge for stochastic coefficients as it is computationally expensive to solve the local spectral problems for every realization of the coefficient. To tackle this computational burden, we propose a machine learning approach. Our method is based on the use of a deep neural network (DNN) to approximate the relation between the stochastic coefficients and the coarse spaces. For the input of the DNN, we apply the Karhunen–Loève expansion and use the first few dominant terms in the expansion. The output of the DNN is the resulting coarse space, which is then applied with the standard adaptive BDDC algorithm. We will present some numerical results with oscillatory and high contrast coefficients to show the efficiency and robustness of the proposed scheme.

Block FETI–DP/BDDC preconditioners for mixed isogeometric discretizations of three-dimensional almost incompressible elasticity

A block FETI–DP/BDDC preconditioner for mixed formulations of almost incompressible elasticity is constructed and analyzed; FETI–DP (Finite Element Tearing and Interconnecting Dual-Primal) and BDDC (Balancing Domain Decomposition by Constraints) are two very successful domain decomposition algorithms for a variety of elliptic problems. The saddle point problems of the mixed problems are discretized with mixed isogeometric analysis with continuous pressure fields. As in previous work by Tu and Li (2015) for finite element discretizations of the incompressible Stokes system, the proposed preconditioner is applied to a reduced positive definite system involving only the pressure interface variable and the Lagrange multiplier of the FETI–DP algorithm. In this work, we extend the theory to a wider class of saddle point problems and we propose a novel block-preconditioning strategy, which consists in using BDDC with deluxe scaling for the interface pressure block as well as deluxe scaling for the FETI–DP preconditioner for the Lagrange multiplier block. A convergence rate analysis is presented with a condition number bound for the preconditioned operator which depends on the inf-sup parameter of the fully assembled problem and the condition number of a closely related BDDC algorithm for compressible elasticity. This bound is scalable in the number of subdomains, poly-logarithmic in the ratio of subdomain and element sizes, and robust with respect to material incompressibility. Parallel numerical experiments validate the theory, demonstrate robustness in the presence of discontinuities of the Lamé parameters, and indicate how the rate of convergence varies with respect to the spline polynomial degree and regularity and the deformation of the domain.