V. Belsky

Multi-grid method for periodic heterogeneous media Part 2: Multiscale modeling and quality control in multidimensional case

Abstract A multi-grid method for a periodic heterogeneous medium in multidimensions is developed. Based on the homogenization theory, special intergrid transfer operators have been developed to stimulate a low frequency response of the boundary value problem with oscillatory coefficients. An adaptive strategy is developed to form a nearly optimal two-scale computational model consisting of the finite element mesh entirely constructed on the microscale in the regions identified by the idealization error indicators, while elsewhere, the modeling level is only sufficient to capture the response of homogenized medium. Numerical experiments show the usefulness of the proposed adaptive multi-level procedure for predicting a detailed response of composite specimens.

Multigrid method for periodic heterogeneous media Part 1: Convergence studies for one-dimensional case

Abstract A multi-grid method for a periodic heterogeneous medium in 1-D is presented. Based on the homogenization theory, special integrid transfer operators have been developed to simulate a low frequency response of the differential equations with oscillatory coefficients. The proposed multi-grid method has been proved to have a fast rate of convergence governed by the ratio q (4− q ) , where 0

Generalized Aggregation Multilevel solver

The paper presents a Generalized Aggregation Multilevel (GAM) solver, w automatically constructs nearly optimal auxiliary coarse models based on the inform available in the source grid only. GAM solver is a hybrid solution scheme wh approximation space of each aggregate (group of neighboring elements) is adaptive automatically selected depending on the spectral characteristics of individual aggre Adaptive features include automated construction of auxiliary aggregated mod tracing “stiff” and “soft” elements, adaptive selection of intergrid transfer operators, adaptive smoothing. An obstacle test consisting of nine industry problems, such as ring-strut-ring stru casting setup in airfoil, nozzle for turbines, turbine blade and diffuser casing as well poor conditioned shell problems, such as High Speed Civil Transport, automobile and canoe, was designed to test the performance of GAM solver. Comparison to th of the art direct and iterative (PCG with Incomplete Cholesky preconditioner) is ca out. Numerical experiments indicate that GAM solver possesses an optimal ra convergence by which the CPU time grows linearly with the problem size, and at the time, robustness is not compromised, as its performance is almost insensitive to pr conditioning. 1.0 Introduction The performance of linear solvers in terms of CPU time for symmetric positive de systems can be approximated as , where N is the number of degrees-of-freedom, a C, β are solution method dependent parameters. The major advantage of direct so their robustness, which is manifested by the fact that parameters C andβ are independent of problem conditioning (except for close to singular systems). Direct solvers are ide solving small up to medium size problems since the constant C for direct methods is significantly smaller than for iterative solvers, but becomes prohibitively expensive for scale problems since the value of exponent for direct solvers is higher than for ite methods. To make direct solvers more efficient various modifications of Gaussian eli CNβ

UNSTRUCTURED MULTIGRID METHOD FOR SHELLS

SUMMARY An accelerated multigrid method, which exploits shell element formulation to speed up the iterative process, is developed for inherently poor conditioned thin domain problems on unstructured grids. Its building blocks are: (i) intergrid transfer operators based on the shell element shape functions, (ii) heavy smoothing procedures in the form of Modified Incomplete Cholesky factorization, and (iii) various two- and threeparameter acceleration schemes. Both the flat shell triangular element and the assumed strain degenerated solid shell element are considered. Numerical results show a remarkable robustness for a wide spectrum of span/thickness ratios encountered in practical applications.

Hierarchical composite grid method for global-local analysis of laminated composite shells

Abstract A hierarchical version of the composite grid method (denoted as HFAC), which exploits the solution of the shell model in studying local effects via a 3D solid model, is developed. Convergence studies on a beam/2D model problem indicate that the spectral radius of the point iteration matrix for the HFAC method is O(1) and O(( L H ) 2 ) with exact and approximate auxiliary coarse grid solutions, respectively, where L and H are the span and the thickness of the beam, respectively. Numerical experiments in multidimensions confirm these findings.

Composite grid method for hybrid systems

Abstract A composite grid method is developed for solving symmetric indefinite linear systems arising from the three-field hybrid variational principle, which weakly enforces compatability and traction continuity conditions between independently modeled substructures. The new solution approach decomposes the hybrid system into a hierarchical global-local problem consisting of a positive definite global problem and an indefinite local system that are derived using minimization on the subspace and the stationarity principle, respectively. An innovative transformation of the indefinite local problem into an equivalent symmetric positive definite system makes the solution of the local problem possible either by a direct solver without pivoting or by a preconditioned iterative method. Alternative semi-iterative solution procedures for the case when auxiliary coarse grid is not available, are also developed. This includes a multigrid-like scheme that uses a mathematical model based on the collocation formulation as the auxiliary grid. Numerical performance studies for two-dimensional plane stress and shell problems are carried out to test the usefulness of the proposed procedures.

Automated adaptive multilevel solver

Abstract This paper presents an automated adaptive multilevel solver for linear (or linearized) system of equations. The multilevel aspect of the solver is aimed at securing an optimal rate of convergence, while keeping the size of the coarsest problem sufficiently small to ensure that the direct portion of the solution does not dominate the total computational cost. Adaptivity in terms of a priori selection of the number of levels (one or more) and construction of the optimal multilevel preconditioner is the key to the robustness of the method. The number of levels is selected on the basis of estimated conditioning, sparsity of the factor and available memory. The auxiliary coarse models (if required) are automatically constructed on the basis of spectral characteristics of individual aggregates (groups of neighboring elements). An obstacle test consisting of twenty industry and model problems was designed to (i) determine the optimal values of computational parameters and to (ii) compare the adaptive multilevel scheme with existing state-of-the-art equation solvers including the Multifrontal solver [17] with the MMD reordering scheme, and the PCG solver with the nearly optimal Modified Incomplete Cholesky factorization preconditioner.

Computer-aided multiscale modeling tools for composite materials and structures☆

Abstract This paper presents recent research efforts at Rensselaer Polytechnic Institute aimed at developing computer-aided multiscale modeling tools for composite materials and structures aimed at predicting the macromechanical (overall) structural response, such as critical deformation, vibration and buckling modes, as well as various failure modes on the mesomechanical (lamina) level, such as delamination and ply buckling, and on the micromechanical (the scale of microconstituents) level, such as debonding, microbuckling, etc. The building blocks of this technology are (i) idealization error estimators aimed at quantifying the quality of the numerical and mathematical models of composites, (ii) multigrid technology aimed at superconvergent solution of the multiscale computational models, (iii) mathematical homogenization theory aimed at constructing inter-scale transfer operators for rapid and reliable information flow between the scales, (iv) system identification for in situ characterization of the phases and their interface, and (v) multiscale model construction and visualization.

Iterative and direct solvers for interface problems with lagrange multipliers

Abstract Special purpose iterative and direct solvers are proposed for solving nonpositive definite symmetric linear systems arising from the three-field hybrid variational principle, which enforces compatibility between independently modeled substructures in the weak sense. The basic idea of the proposed family of solvers is to transform a nonpositive definite linear system into an equivalent positive definite system, which can then be solved by either the iterative or direct method without pivoting. The two-level iterative approach proposed in this paper consists of resolving the lower frequency response of the source problem by means of the equivalent positive definite collocation problem and capturing higher frequency response using projected conjugate gradient method. Numerical experiments in 2-D and shells indicate that the total CPU time of the two-level iterative process is less than 20% higher than that of a corresponding collocation problem, but the benefit from increased accuracy and modeling flexibility clearly overshadows this increased cost.

Efficient solutions schemes for interface problems

Abstract A computationally efficient two-level iterative scheme is proposed for the solution of the interface problems with Lagrange multipliers, where the oscillatory part of the solution is resolved by means off smoothing using a new, efficient preconditioner whereas the smooth component of the solution is captured by the collocation-based problem on the auxilliary grid, that is solved directly using a sparse direct solver. A simple adaptive feature is built into the proposed solution method in order to guarantee convergence for ill-conditioned problems. Nmerical results presented for example problems including that of a Boeing crown panel show that the proposed tww-level solution technique outperfrmsnce the standard, single level iterative and direect solvers.