Milan D. Mihajlovic

An efficient preconditioner for monolithically-coupled large-displacement fluid-structure interaction problems with pseudo-solid mesh updates

We present a block preconditioner for the efficient solution of the linear systems that arise when employing Newtons method to solve monolithically-coupled large-displacement fluid-structure interaction problems in which the update of the moving fluid mesh is performed by the equations of large-displacement elasticity. Following a theoretical analysis of the preconditioner, we propose an efficient implementation that yields a solver with near-optimal computational cost, in the sense that the time for the solution of the linear systems scales approximately linearly with the number of unknowns. We evaluate the performance of the preconditioner in selected two- and three-dimensional test problems.

Fast iterative solvers for buoyancy driven flow problems

We outline a new class of robust and efficient methods for solving the Navier-Stokes equations with a Boussinesq model for buoyancy driven flow. We describe a general solution strategy that has two basic building blocks: an implicit time integrator using a stabilized trapezoid rule with an explicit Adams-Bashforth method for error control, and a robust Krylov subspace solver for the spatially discretized system. We present numerical experiments illustrating the efficiency of the chosen preconditioning schemes with respect to the discretization parameters.

A numerical investigation of the solution of a class of fourth–order eigenvalue problems

This paper is concerned with the accurate numerical approximation of the spectral properties of the biharmonic operator on various domains in two dimensions. A number of analytic results concerning the eigenfunctions of this operator are summarized and their implications for numerical approximation are discussed. In particular, the asymptotic behaviour of the first eigenfunction is studied since it is known that this has an unbounded number of oscillations when approaching certain types of corners on domain boundaries. Recent computational results of Bjørstad & Tjøstheim, using a highly accurate spectral Legendre–Galerkin method, have demonstrated that a number of these sign changes may be accurately computed on a square domain provided sufficient care is taken with the numerical method. We demonstrate that similar accuracy is also achieved using an unstructured finite–element solver which may be applied to problems on domains with arbitrary geometries. A number of results obtained from this mixed finite–element approach are then presented for a variety of domains. These include a family of circular sector regions, for which the oscillatory behaviour is studied as a function of the internal angle, and another family of (symmetric and non–convex) domains, for which the parity of the least eigenfunction is investigated. The paper not only verifies existing asymptotic theory, but also allows us to make a new conjecture concerning the eigenfunctions of the biharmonic operator.

Efficient parallel solvers for the biharmonic equation

Abstract We examine the convergence characteristics and performance of parallelised Krylov subspace solvers applied to the linear algebraic systems that arise from low-order mixed finite element approximation of the biharmonic problem. Our strategy results in preconditioned systems that have nearly optimal eigenvalue distribution, which consists of a tightly clustered set together with a small number of outliers. We implement the preconditioner operator in a “black-box” fashion using publicly available parallelised sparse direct solvers and multigrid solvers for the discrete Dirichlet Laplacian. We present convergence and timing results that demonstrate efficiency and scalability of our strategy when implemented on contemporary computer architectures.

A component decomposition preconditioning for 3D stress analysis problems

A preconditioning methodology for an iterative solution of discrete stress analysis problems based on a space decomposition and subspace correction framework is analysed in this paper. The principle idea of our approach is a decomposition of a global discrete system into the series of subproblems each of which correspond to the different Cartesian co-ordinates of the solution (displacement) vector. This enables us to treat the matrix subproblems in a segregated way. A host of well-established scalar solvers can be employed for the solution of subproblems. In this paper we constrain ourselves to an approximate solution using the scalar algebraic multigrid (AMG) solver, while the subspace correction is performed either in block diagonal (Jacobi) or block lower triangular (Gauss–Seidel) fashion. The preconditioning methodology is justified theoretically for the case of the block-diagonal preconditioner using Korns inequality for estimating the ratio between the extremal eigenvalues of a preconditioned matrix. The effectiveness of the AMG-based preconditioner is tested on stress analysis 3D model problems that arise in microfabrication technology. The numerical results, which are in accordance with theoretical predictions, clearly demonstrate the superiority of a component decomposition AMG preconditioner over the standard ILU preconditioner, even for the problems with a relatively small number of degrees of freedom. Copyright

An efficient direct solver for a class of mixed finite element problems

In this paper we present an efficient, accurate and parallelizable direct method for the solution of the (indefinite) linear algebraic systems that arise in the solution of fourth-order partial differential equations (PDEs) using mixed finite element approximations. The method is intended particularly for use when multiple right-hand sides occur, and when high accuracy is required in these solutions. The algorithm is described in some detail and its performance is illustrated through the numerical solution of a biharmonic eigenvalue problem where the smallest eigenpair is approximated using inverse iteration after discretization via the Ciarlet–Raviart mixed finite element method.

EFFICIENT BLOCK PRECONDITIONING FOR A C 1 FINITE ELEMENT DISCRETIZATION OF THE DIRICHLET BIHARMONIC PROBLEM

We present an efficient block preconditioner for the two-dimensional biharmonic Dirichlet problem discretised by C1 bicubic Hermite finite elements. In this formulation each node in the mesh has four different degrees of freedom (DOFs). Grouping DOFs of the same type together leads to a natural blocking of the Galerkin coefficient matrix. Based on this block structure, we develop two preconditioners: a 2x2 block diagonal preconditioner (BD) and a block bordered diagonal (BBD) preconditioner. We prove mesh independent bounds for the spectra of the BD-preconditioned Galerkin matrix under certain conditions. The eigenvalue analysis is based on the fact that the proposed preconditioner, like the coefficient matrix itself, is symmetric positive definite and is assembled from element matrices. We demonstrate the effectiveness of an inexact version of the BBD preconditioner, which exhibits near optimal scaling in terms of computational cost with respect to the discrete problem size. Finally, we study robustness of this preconditioner with respect to element stretching, domain distortion and non-convex domains.

IC thermal analyzer for versatile 3-D structures using multigrid preconditioned krylov methods

Thermal analysis is crucial for determining the propagation of heat and tracking the formation of hot spots in advanced integrated circuit technologies. At the core of the thermal analysis for integrated circuits is the numerical solution of the heat equation. Prior academic thermal analysis tools typically compute temperature by applying finite difference methods on uniform grids with time integration methods having fixed time step size. Additionally, the linear systems arising from the discretized heat equation are solved using direct methods based on matrix factorizations. Direct methods, however, do not scale well as the problem size increases. Moreover, most of the tools support only 2-D or a limited number of 3-D technologies. To address these issues, this paper presents a novel thermal analyzer with the ability to model both 2-D and 3-D circuit technologies. The analyzer solves the heat equation using the finite element method for the spatial discretization coupled with implicit time integration methods for advancing the solution in time. It also offers fully adaptive spatio-temporal refinement features for improved accuracy and computational efficiency. The resulting linear systems are solved by a multigrid preconditioned Krylov subspace iterative method, which gives superior performance for 3-D transient analyses. The analyzer is shown to accurately capture the propagation of heat in both the horizontal and vertical directions of integrated systems.

Efficiency Study of the “Black-Box” Component Decomposition Preconditioning for Discrete Stress Analysis Problems

Efficiency of the preconditioning methodology for an iterative solution of discrete stress analysis problems is studied in this article. The preconditioning strategy is based on space decomposition and subspace correction framework. The principle idea is to decompose a global discrete system into a sequence of scalar subproblems, corresponding to the different Cartesian coordinates of the displacement vector. The scalar subproblems can be treated by a host of direct and iterative techniques, however we restrict ourselves to a “black-box” application of the direct sparse solvers and the scalar algebraic multigrid (AMG) method. The subspace correction is performed in either block diagonal or block lower triangular fashion. The efficiency and potential limitations of the proposed preconditioning methodology are studied on stress analysis for 2D and 3D model problems from microfabrication technology.

The MTA: An Advanced and Versatile Thermal Simulator for Integrated Systems

Fast and accurate thermal analysis is crucial for determining the propagation of heat and tracking the formation of hotspots in integrated circuits (ICs). Existing academic thermal analysis tools primarily use compact models to accelerate thermal simulations but are limited to linear problems on relatively simple circuit geometries. The Manchester Thermal Analyzer (MTA) is a comprehensive tool that allows for fast and highly accurate linear and nonlinear thermal simulations of complex physical structures including the IC, the package, and the heatsink. The MTA is targeted for 2.5/3-D IC designs but also handles standard planar ICs. The MTA discretizes the heat equation in space using the finite element method and performs the time integration with unconditionally stable implicit time stepping methods. To improve the computational efficiency without sacrificing accuracy, the MTA features adaptive spatiotemporal refinement. The large-scale linear systems that arise during the simulation are solved with fast preconditioned Krylov subspace methods. The MTA supports thermal analysis of realistic integrated systems and surpasses the computational capabilities and performance of existing academic thermal simulators. For example, the simulation of a processor in a package attached to a heat sink, modeled by a computational grid consisting of over 3 million nodes, takes less than 3 min. The MTA is fully parallel and publicly available.1