A TFETI Domain Decomposition Solver for Elastoplastic Problems

We propose an algorithm for the efficient parallel implementation of elastoplastic problems with hardening based on the so-called TFETI (Total Finite Element Tearing and Interconnecting) domain decomposition method. We consider an associated elastoplastic model with the von Mises plastic criterion and the linear isotropic hardening law. Such a model is discretized by the implicit Euler method in time and the consequent one time step elastoplastic problem by the finite element method in space. The latter results in a system of nonlinear equations with a strongly semismooth and strongly monotone operator. The semismooth Newton method is applied to solve this nonlinear system. Corresponding linearized problems arising in the Newton iterations are solved in parallel by the above mentioned TFETI domain decomposition method. The proposed TFETI based algorithm was implemented in Matlab parallel environment and its performance was illustrated on a 3D elastoplastic benchmark. Numerical results for different time discretizations and mesh levels are presented and discussed and a local quadratic convergence of the semismooth Newton method is observed

Matlab parallel codes for 3D slope stability benchmarks

This contribution is focused on a description of implementation details for solver related to the slope stability benchmarks in 3D. Such problems are formulated by the standard elastoplastic models containing the Mohr-Coulomb yield criterion and by the limit analysis of collapse states. The implicit Euler method and higher order finite elements are used for discretization. The discretized problem is solved by non-smooth Newton-like methods in combination with incremental methods of limit load analysis. In this standard approach, we propose several innovative techniques. Firstly, we use recently developed sub-differential based constitutive solution schemes. Such an approach is suitable for non-smooth yield criteria, and leads better return-mapping algorithms. For example, a priori decision criteria for each return-type or simplified construction of consistent tangent operators are applied. The parallel codes are developed in MATLAB using Parallel Computing Toolbox. For parallel implementation of linear systems, we use the TFETI domain decomposition method. It is a non-overlapping method where the Lagrange multipliers are used to enforce continuity on the subdomain interfaces and satisfaction of the Dirichlet boundary conditions

A strict error bound with separated contributions of the discretization and of the iterative solver in non-overlapping domain decomposition methods

This paper deals with the estimation of the distance between the solution of a static linear mechanic problem and its approximation by the finite element method solved with a non-overlapping domain decomposition method (FETI or BDD). We propose a new strict upper bound of the error which separates the contribution of the iterative solver and the contribution of the discretization. Numerical assessments show that the bound is sharp and enables us to define an objective stopping criterion for the iterative solverComment: Computer Methods in Applied Mechanics and Engineering (2013) onlin

Adaptive Coarse Spaces for FETI-DP and BDDC Methods

Iterative substructuring methods are well suited for the parallel iterative solution of elliptic partial differential equations. These methods are based on subdividing the computational domain into smaller nonoverlapping subdomains and solving smaller problems on these subdomains. The solutions are then joined to a global solution in an iterative process. In case of a scalar diffusion equation or the equations of linear elasticity with a diffusion coefficient or Young modulus, respectively, constant on each subdomain, the numerical scalability of iterative substructuring methods can be proven. However, the convergence rate deteriorates significantly if the coefficient in the underlying partial differential equation (PDE) has a high contrast across and along the interface of the substructures. Even sophisticated scalings often do not lead to a good convergence rate. One possibility to enhance the convergence rate is to choose appropriate primal constraints. In the present work three different adaptive approaches to compute suitable primal constraints are discussed. First, we discuss an adaptive approach introduced by Dohrmann and Pechstein that draws on the operator P_D which is an important ingredient in the analysis of iterative substructuring methods like the dual-primal Finite Element Tearing and Interconnecting (FETI-DP) method and the closely related Balancing Domain Decomposition by Constraints (BDDC) method. We will also discuss variations of the method by Dohrmann and Pechstein introduced by Klawonn, Radtke, and Rheinbach. Secondly, we describe an adaptive method introduced by Mandel and Sousedík which is also based on the P_D-operator. Recently, a proof for the condition number bound in this method was provided by Klawonn, Radtke, and Rheinbach. Thirdly, we discuss an adaptive approach introduced by Klawonn, Radtke, and Rheinbach that enforces a Poincaré- or Korn-like inequality and an extension theorem. In all approaches generalized eigenvalue problems are used to compute a coarse space that leads to an upper bound of the condition number which is independent of the jumps in the coefficient and depend on an a priori prescribed tolerance. Proofs and numerical tests for all approaches are given in two dimensions. Finally, all approaches are compared

Parallel harmonic balance method for analysis of nonlinear mechanical systems

Mechanical vibration analysis and modelling are essential tools used in the design of various mechanical components and structures. In the case of turbine engine design specifically, the ability to accurately predict vibration of various parts is crucial to ensure their safe operation while maintaining efficiency. As the designs become increasingly complex and margins for errors get smaller, high fidelity numerical vibration models are necessary for their analysis. Research of parallel algorithms has progressed significantly in the last decades, thanks to the exponential growth of the world's available computational resources. This work explores the possibilities for parallel implementations for solving large scale nonlinear vibration problems. A C++ code using MPI was developed to validate these implementations in practice. The harmonic balance method is used in combination with finite elements discretisation and applied to an elastic body with the Green-Lagrange nonlinear model for large deformations. A parameter continuation scheme using a predictor-corrector approach is included to compute frequency response functions. A Newton-Raphson solver is used to solve the bordered nonlinear system of equations in the frequency domain. Three different parallel algorithms for solving the linearised problem in each Newton iteration are analysed - a sparse direct solver (using MUMPS library), GMRES (using PETSc library) and an inhouse implementation of FETI. The performance of the solvers is analysed using beam testcases and a fan blade geometry. Scalability of MUMPS and the FETI solver is assessed. Full nonlinear frequency response functions with turning points are also computed. Use of artificial coarse space and preconditioning in FETI is discussed as it greatly impacts convergence properties of the solver. The presented parallel linear solvers show promising scalability results and an ability to solve nonlinear systems of several million degrees of freedom.Open Acces

Advanced Computational Engineering

The finite element method is the established simulation tool for the numerical solution of partial differential equations in many engineering problems with many mathematical developments such as mixed finite element methods (FEMs) and other nonstandard FEMs like least-squares, nonconforming, and discontinuous Galerkin (dG) FEMs. Various aspects on this plus related topics ranging from order-reduction methods to isogeometric analysis has been discussed amongst the pariticpants form mathematics and engineering for a large range of applications

Parallel computation in efficient non-linear finite element analysis with applications to soft-ground tunneling project

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, 2004.Includes bibliographical references.Reliable prediction and control of ground movements represent an essential component of underground construction projects in congested urban environments, to mitigate against possible damage to adjacent structures and utilities. This research was motivated by the construction of a large underground cavern for the Rio Piedras station in San Juan, Puerto Rico. This project involved the construction of a large, horseshoe-shaped cavern (17m wide and 16m high) in weathered alluvial soils. The crown of the cavern is located less than 5.5m below existing buildings in a busy commercial district. Structural support for the cavern was provided by a series of 15 stacked drifts. These 3m square-section galleries were excavated mainly by hand and in-filled with concrete, while a compensation grouting system was designed to mitigate effects of excavation-induced ground movements on the overlying structures. Unexpectedly large settlements occurred during drift construction and overwhelmed the grouting system that was intended to compensate for tunnel-induced movements. Although two dimensional, non-linear finite element analyses of the stacked- drift construction suggest that movements exceeding 100mm can be expected, the 2-D representation of excavation and ground support is overly simplistic and represents a major source of uncertainty in these analyses. Massive computational efforts make more comprehensive 3-D models of the construction sequence completely impractical using existing finite element software with direct or iterative solver methods.(cont.) This thesis develops, implements, and applies an efficient parallel computation scheme for solving such large-scale, non-linear finite element analyses. The analyses couple a non- overlapping Domain Decomposition technique known as the FETI algorithm (Farhat & Roux, 1991) with a Newton-Raphson iteration scheme for non-linear material behavior. This method uses direct factorization of the equilibrium equations for sub-domains, while solving a separate interface problem iteratively with a mechanically consistent, Dirichlet pre- conditioner. The implementation allows independence of the number of sub-domains from the number of processors. This provides flexibility on mesh decomposition, control between iterative interface solutions and direct sub-domain solutions, and load balance in shared heterogeneous clusters. The analyses are performed with the developed code, FETI- FEM (programmed in C++ and MPI) using syntax consistent with pre-existing ABAQUS software. Benchmark testing on a Beowulf cluster of 16 interconnected commodity PC computers found excellent parallel efficiency, while the computation time scales with the number of finite elements, NE, according to a power law with exponent, p = 1.217. Parallel 3-D FE analyses have been applied in modeling the drift excavation, primary lining and infilling for the stacked-drift construction assuming a simplified soil profile. The resulting FE model comprised approximately 30,000 20-noded quadratic displacement-based elements, representing almost 400,000 degrees of freedom (at least one order of magnitude larger than any prior model reported in the geotechnical literature) and was sub-divided into 168 sub-domains ...by Yo-Ming Hsieh.Ph.D

High-Performance Computing Two-Scale Finite Element Simulations of a Contact Problem Using Computational Homogenization - Virtual Forming Limit Curves for Dual-Phase Steel

The appreciated macroscopic properties of dual-phase (DP) steels strongly depend on their microstructure. Therefore, accurate finite element (FE) simulations of a deformation process of such a steel require the incorporation of the microscopic heterogeneous structure. Usually, a brute force FE discretization incorporating the microstructure is not feasible since it results in exceedingly large problem sizes. Instead, the microstructure has to be incorporated by using computational homogenization. We present a numerical two-scale approach of the Nakajima test for a DP steel, which is a well known material test in the steel industry. It can be used to derive forming limit diagrams (FLDs), which allow experts to judge the maximum formability properties of a specific type of sheet metal in the considered thickness. For the simulations, we use our software package FE2TI, which is a highly scalable implementation of the well known FE2 homogenization approach. The microstructure is represented by a representative volume element (RVE) and it is discretized separately from the macroscopic problem. We discuss the incorporation of contact constraints using a penalty formulation as well as appropriate boundary conditions. In addition, we introduce a simple load step strategy and different opportunities for the choice of an initial value for a single load step by using an interpolation polynomial. Finally, we come up with computationally derived FLDs. Although we use a computational homogenization strategy, the resulting problems on both scales can be quite large. The efficient solution of such large problems requires parallel strategies. Therefore, we consider the highly scalable nonlinear domain decomposition methods FETI-DP (Finite Element Tearing and Interconnecting - Dual-Primal) and BDDC (Balancing Domain Decomposition by Constraints). For the first time, the BDDC approach is used for the parallel solution of the macroscopic problem in a simulation of the Nakajima test. We introduce a unified framework that combines all variants of nonlinear FETI-DP and nonlinear BDDC. For the first time, we introduce a nonlinear FETI-DP variant that chooses suitable elimination sets by utilizing information from the nonlinear residual. Furthermore, we show weak scaling results for different nonlinear FETI-DP variants and several model problems

Efficient Residual and Matrix-free Jacobian Evaluation for Three-Dimensional Hexahedral Finite Elements with Nearly-Incompressible Neo-Hookean Hyperelasticity as Applied to Soft Materials

We examine a residual and matrix-free Jacobian formulation of compressible and nearly incompressible (&nu;&rarr; 0.5) displacement-only linear isotropic elasticity with high-order hexahedral finite elements. In addition, we consider two formulations for the matrix-free action of the Jacobian in Neo-Hookean hyperelasticity at finite strain: (i) working in the reference configuration to define the second Piola-Kirchhoff tensor as a function of the Green-Lagrange strain (or equivalently, the right Cauchy-Green tensor), and (ii) working in the current configuration to define the Kirchhoff stress in terms of the left Cauchy-Green tensor. We explore accuracy-time-cost tradeoffs for the prediction of strain energy across the range of polynomial degrees and Jacobian representations. Finally, we consider soft materials such as rubber, elastomers, and soft biological tissues. Simulating these large isochoric deformations computationally, such as with Finite Element Analysis (FEA), requires higher order (typically quadratic) interpolation functions and/or enhancements through hybrid/mixed methods to maintain stability. Lower order (linear) finite elements with hybrid/mixed formulation may not perform stably for all mechanical loading scenarios involving large isochoric deformations, whereas quadratic finite elements with or without hybrid/mixed formulation may perform stably, especially when large bending or folding deformations are being simulated. For topology-optimization design of soft robotics, for instance, the FEA solid mechanics solver must run efficiently and stably. Stability is ensured by a higher order finite element formulation (with or without enhancements), but efficiency for higher order FEA remains a concern. Thus, this thesis addresses the efficiency concern from the perspective of computer science algorithms and programming. In all cases, $p$-multigrid method is combined with algebraic multigrid on the assembled sparse coarse grid matrix to provide an effective preconditioner.</p

Computational Engineering

The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods