207 research outputs found

    Constraint interface preconditioning for topology optimization problems

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    The discretization of constrained nonlinear optimization problems arising in the field of topology optimization yields algebraic systems which are challenging to solve in practice, due to pathological ill-conditioning, strong nonlinearity and size. In this work we propose a methodology which brings together existing fast algorithms, namely, interior-point for the optimization problem and a novel substructuring domain decomposition method for the ensuing large-scale linear systems. The main contribution is the choice of interface preconditioner which allows for the acceleration of the domain decomposition method, leading to performance independent of problem size.Comment: To be published in SIAM J. Sci. Com

    Computational Engineering

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    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

    The INTERNODES method for applications in contact mechanics and dedicated preconditioning techniques

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    The mortar finite element method is a well-established method for the numerical solution of partial differential equations on domains displaying non-conforming interfaces. The method is known for its application in computational contact mechanics. However, its implementation remains challenging as it relies on geometrical projections and unconventional quadrature rules. The INTERNODES (INTERpolation for NOn-conforming DEcompositionS) method, instead, could overcome the implementation difficulties thanks to flexible interpolation techniques. Moreover, it was shown to be at least as accurate as the mortar method making it a very promising alternative for solving problems in contact mechanics. Unfortunately, in such situations the method requires solving a sequence of ill-conditioned linear systems. In this paper, preconditioning techniques are designed and implemented for the efficient solution of those linear systems

    Partitioning strategies for the interaction of a fluid with a poroelastic material based on a Nitsche's coupling approach

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    We develop a computational model to study the interaction of a fluid with a poroelastic material. The coupling of Stokes and Biot equations represents a prototype problem for these phenomena, which feature multiple facets. On one hand it shares common traits with fluid-structure interaction. On the other hand it resembles the Stokes-Darcy coupling. For these reasons, the numerical simulation of the Stokes-Biot coupled system is a challenging task. The need of large memory storage and the difficulty to characterize appropriate solvers and related preconditioners are typical shortcomings of classical discretization methods applied to this problem. The application of loosely coupled time advancing schemes mitigates these issues because it allows to solve each equation of the system independently with respect to the others. In this work we develop and thoroughly analyze a loosely coupled scheme for Stokes-Biot equations. The scheme is based on Nitsche's method for enforcing interface conditions. Once the interface operators corresponding to the interface conditions have been defined, time lagging allows us to build up a loosely coupled scheme with good stability properties. The stability of the scheme is guaranteed provided that appropriate stabilization operators are introduced into the variational formulation of each subproblem. The error of the resulting method is also analyzed, showing that splitting the equations pollutes the optimal approximation properties of the underlying discretization schemes. In order to restore good approximation properties, while maintaining the computational efficiency of the loosely coupled approach, we consider the application of the loosely coupled scheme as a preconditioner for the monolithic approach. Both theoretical insight and numerical results confirm that this is a promising way to develop efficient solvers for the problem at hand

    A Computational Framework for Axisymmetric Linear Elasticity and Parallel Iterative Solvers for Two-Phase Navier–Stokes

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    This dissertation explores ways to improve the computational efficiency of linear elasticity and the variable density/viscosity Navier--Stokes equations. While the approaches explored for these two problems are much different in nature, the end goal is the same - to reduce the computational effort required to form reliable numerical approximations.\\ The first topic considered is the axisymmetric linear elasticity problem. While the linear elasticity problem has been studied extensively in the finite-element literature, to the author\u27s knowledge, this is the first study of the elasticity problem in an axisymmetric setting. Indeed, the axisymmetric nature of the problem means that a change of variables to cylindrical coordinates reduces a three-dimensional problem into a decoupled one-dimensional and two-dimensional problem. The change of variables to cylindrical coordinates, however, affects the functional form of the divergence operator and the definition of the inner products. To develop a computational framework for the linear elasticity problem in this context, a new projection operator is defined that is tailored to the cylindrical form of the divergence and inner products. Using this framework, a stable finite-element quadruple is derived for k=1,2k=1,2. These computational rates are then validated with a few computational examples.\\ The second topic addressed in this work is the development of a new Schur complement approach for preconditioning the two-phase Navier--Stokes equations. Considerable research effort has been invested in the development of Schur complement preconditioning techniques for the Navier--Stokes equations, with the pressure-convection diffusion (PCD) operator and the least-squares commutator being among the most popular. Furthermore, more recently researchers have begun examining preconditioning strategies for variable density / viscosity Stokes and Navier--Stokes equations. This work contributes to recent work that has extended the PCD Schur complement approach for single phase flow to the variable phase case. Specifically, this work studies the effectiveness of a new two-phase PCD operator when applied to dynamic two-phase simulations that use the two-phase Navier--Stokes equations. To demonstrate the new two-phase PCD operators effectiveness, results are presented for standard benchmark problems, as well as parallel scaling results are presented for large-scale dynamic simulations for three-dimensional problems
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