Zdeněk Johan

A new finite element formulation for computational fluid dynamics: X. The compressible Euler and Navier-Stokes equations

A space-time element method is presented for solving the compressible Euler and Navier-Stokes equations. The proposed formulation includes the variational equation, predictor multi-corrector algorithms and boundary conditions. The variational equation is based on the time-discontinuous Galerkin method, in which the physical entropy variables are employed. A least-squares operator and a discontinuity-capturing operator are added, resulting in a high-order accurate and unconditionally stable method. Implicit/explicit predictor multi-corrector algorithms, applicable to steady as well as unsteady problems, are presented; techniques are developed to enhance their efficiency. Implementation of boundary conditions is addressed; in particular, a technique is introduced to satisfy nonlinear essential boundary conditions, and a consistent method is presented to calculate boundary fluxes. Numerical results are presented to demonstrate the performance of the method.

A globally convergent matrix-free algorithm for implicit time-marching schemes arising in finite element analysis in fluids

A solution procedure for solving nonlinear time-marching problems is presented. The nonsymmetric systems of equations arising from a Newton-type linearization of these time-marching problems are solved using an iterative strategy based on the generalized minimal residual (GMRES) algorithm. Matrix-free techniques leading to reduction in storage are presented. Incorporation of a linesearch algorithm in the Newton-GMRES scheme is discussed. An automatic time-increment control strategy is developed to increase the stability of the time-marching process. High-speed flow computations demonstrate the effectiveness of these algorithms.

A multi-element group preconditioned GMRES algorithm for nonsymmetric systems arising in finite element analysis

Abstract A multi-element group, domain decomposition algorithm is presented for solving linear nonsymmetric systems arising in finite element analysis. The iterative strategy employed is based on the generalized minimum residual (GMRES) procedure originally proposed by Saad and Shultz. Two levels of preconditioning are investigated. Coding is presented which fully exploits vector-architectured computers. Applications to problems of high-speed compressible flow illustrate the effectiveness of the scheme.

A data parallel finite element method for computational fluid dynamics on the Connection Machine system

Abstract A finite element method for computational fluid dynamics has been implemented on the Connection Machine systems CM-2 and CM-200. An implicit iterative solution strategy, based on the preconditioned matrix-free GMRES algorithm, is employed. Parallel data structures built on both nodal and elemental sets are used to achieve maximum parallelization. Communication primitives provided through the Connection Machine Scientific Software Library substantially improved the overall performance of the program. Computations of three-dimensional compressible flows using unstructured meshes having close to one million elements, such as a complete airplane, demonstrate that the Connection Machine systems are suitable for these applications. Performance comparisons are also carried out with the vector computers Cray Y-MP and Convex C-1.

An efficient communications strategy for finite element methods on the Connection Machine CM-5 system

Abstract The objective of this paper is to propose communication procedures suitable for unstructured finite element solvers implemented on distributed-memory parallel computers such as the Connection Machine CM-5 system. First, a data-parallel implementation of the recursive spectral bisection (RSB) algorithm proposed by Pothen et al. is presented. The RSB algorithm is associated with a node renumbering scheme which improves data locality of reference. Two-step gather and scatter operations taking advantage of this data locality are then designed. These communication primitives make use of the indirect addressing capability of the CM-5 vector units to achieve high gather and scatter bandwidths. The performance of the proposed communication strategy is illustrated on large-scale three-dimensional fluid dynamics problems.

Multiphysics simulation of flow-induced vibrations and aeroelasticity on parallel computing platforms

Abstract This article describes the application of multiphysics simulation on parallel computing platforms to model aeroelastic instabilities and flow-induced vibrations. Multiphysics simulation is based on a single computational framework for the modeling of multiple interacting physical phenomena. Within the multiphysics framework, the finite element treatment of fluids is based on the Galerkin-Least-Squares (GLS) method with discontinuity capturing operators. The arbitrary-Lagrangian—Eulerian (ALE) method is utilized to account for deformable fluid domains. The finite element treatment of solids and structures is based on a three-field variational principle. Fully-coupled interaction constraints are enforced using the augmented-Lagrangian method. The multiphysics architecture lends itself naturally to high-performance parallel computing. Several applications are presented. The importance of capturing the nonlinear effects and accounting for mesh-movement is highlighted and the scalability of the software is illustrated.

Implementation of a one-equation turbulence model within a stabilized finite element formulation of a symmetric advective-diffusive system☆

The success of stabilized finite element methods for symmetric advective-diffusive formulations of the laminar Navier-Stokes equations has motivated extension of these concepts to turbulent flows. A one-equation model which preserves the symmetry of the systems advective and diffusive matrices has been developed. This model is based on the work of Norris and Reynolds and utilizes a transport equation for the turbulent velocity scale q and an algebraic relation for the turbulent length scale l. From these two quantities, an eddy viscosity can be calculated. This model has been implemented by way of a Galerkin/least-squares finite element method. Solutions are presented which verify the good behavior of the formulation.

Scalability of Finite Element Applications on Distributed–Memory Parallel Computers

This paper demonstrates that scalability and competitive efficiency can be achieved for unstructured grid finite element applications on distributed memory machines, such as the Connection Machine CM-5 system. The efficiency of finite element solvers is analyzed through two applications: an implicit computational aerodynamics application and an explicit solid mechanics application. Scalability of mesh decomposition and of data mapping strategies is also discussed. Numerical examples that support the claims for problems with an excess of fourteen million variables are presented.

In-cylinder cold flow simulation using a finite element method

Abstract This paper presents a numerical strategy based on the finite element method for solving unsteady flows without combustion inside an operating engine cylinder. A compressible flow solver that can withstand highly distorted elements is presented. A robust mesh deformation procedure capable of handling the large valve and piston motions and deforming the unstructured mesh accordingly is then described. Turbulence is added through a one-equation model. The proposed strategy is applied to a three-dimensional realistic problem. Results are presented for the exhaust, intake and compression strokes of the cylinder operation.

Automotive design applications of fluid flow simulation on parallel computing platforms

Abstract This article describes the use of fluid flow simulation on parallel computing platforms to solve design problems in the automotive industry. The fluid flow formulation in the Spectrum TM solver, which is used for these simulations, is described. The finite element treatment of fluid flow in this work is based on the Galerkin-Least-Squares (GLS) method with discontinuity capturing operators. The Arbitrary-Lagrangian–Eulerian (ALE) method is utilized to account for deformable fluid domains. Automatically generated tetrahedral grids are used to ease and expedite the analysis process. The multiphysics architecture used in this work lends itself naturally to high-performance parallel computing. By taking advantage of automatic mesh generation and parallel computing, dramatic reduction in turnaround time for flow analysis is achieved. Several applications are presented which demonstrate the utility and accuracy of finite element solutions in automotive engineering problems and highlight the scalability of the software.