Luis Crivelli

Extending substructure based iterative solvers to multiple load and repeated analyses

Direct solvers currently dominate commercial finite element structural software, but do not scale well in the fine granularity regime targeted by emerging parallel processors. Substructure based iterative solvers—often called also domain decomposition algorithms—lend themselves better to parallel processing, but must overcome several obstacles before earning their place in general purpose structural analysis programs. One such obstacle is the solution of systems with many or repeated right hand sides. Such systems arise, for example, in multiple load static analyses and in implicit linear dynamics computations. Direct solvers are well-suited for these problems because after the system matrix has been factored, the multiple or repeated solutions can be obtained through relatively inexpensive forward and backward substitutions. On the other hand, iterative solvers in general are ill-suited for these problems because they often must restart from scratch for every different right hand side. In this paper, we present a methodology for extending the range of applications of domain decomposition methods to problems with multiple or repeated right hand sides. Basically, we formulate the overall problem as a series of minimization problems over K-orthogonal and supplementary subspaces, and tailor the preconditioned conjugate gradient algorithm to solve them efficiently. The resulting solution method is scalable, whereas direct factorization schemes and forward and backward substitution algorithms are not. We illustrate the proposed methodology with the solution of static and dynamic structural problems, and highlight its potential to outperform forward and backward substitutions on parallel computers. As an example, we show that for a linear structural dynamics problem with 11640 degrees of freedom, every time-step beyond time-step 15 is solved in a single iteration and consumes 1.0 second on a 32 processor iPSC-860 system; for the same problem and the same parallel processor, a pair of forward/backward substitutions at each step consumes 15.0 seconds.

A general approach to nonlinear FE computations on shared-memory multiprocessors

Abstract A computational strategy for nonlinear finite element computations on shared-memory multiprocessors is presented. It exploits all the parallelism inherent in the finite element method. Both iterative and direct solution methods are implemented in the same environment. Explicit computations are carried out at the element level. Implicit computations are processed at the subdomain level. An element coloring scheme is used to eliminate critical regions for the whole class of explicit computations. To facilitate load balancing among the processors in a posteriori nonlinear problems, a dynamic remapping of the processors on the computational tasks is introduced. Numerical experiments are conducted on Alliant FX/8 and Cray2. Very high rates of efficiency are achieved on both multiprocessors.

Implicit time integration of a class of constrained hybrid formulations—Part I: Spectral stability theory

Abstract Incomplete field formulations have recently been the subject of intense research because of their potential in coupled analysis of independently modeled substructures, adaptive refinement, domain decomposition and parallel processing. This paper presents a spectral stability theory for the differential/algebraic dynamic systems associated with these formulations, discusses the design and analysis of suitable time-integration algorithms, and emphasizes the treatment of the inter-subdomain linear constraint equations. These constraints are shown to introduce a destabilizing effect in the dynamic system that can be analyzed by investigating the behavior of the time-integration algorithm at infinite and zero frequencies. Three different approaches for constructing penalty-free unconditionally stable second-order accurate solution procedures for this class of hybrid formulations are presented, analyzed and illustrated with numerical examples. In particular, the advantages of the Hilber-Hughes-Taylor (HHT) method and its generalized version (Generalized a) are highlighted. The family of problems discussed in this paper can also be viewed as model problems for the more general case of hybrid formulations with non-linear constraints. For example, it is shown numerically in this paper that the theoretical results predicted by the spectral stability theory also apply to non-linear multibody dynamics formulations. Therefore, some of the algorithms outlined in this work are important alternatives to the popular technique consisting of transforming differential/algebraic equations into ordinary differential equations via the introduction of a stabilization term that depends on arbitrary constants, and that influences the computed solution.

A survey of the core-congruential formulation for geometrically nonlinear TL finite elements

SummaryThis article presents a survey of the Core-Congruential Formulation (CCF) for geometrically nonlinear mechanical finite elements based on the Total Lagrangian (TL) kinematic description. Although the key ideas behind the CCF can be traced back to Rajasekaran and Murray in 1973, it has not subsequently received serious attention. The CCF is distinguished by a two-phase development of the finite element stiffness equations. The initial phase develop equations for individual particles. These equations are expressed in terms of displacement gradients as degrees of freedom. The second phase involves congruential-type transformations that eventually binds the element particles of an individual element in terms of its node-displacement degrees of freedom. Two versions of the CCF, labeled Direct and Generalized, are distinguished. The Direct CCF (DCCF) is first described in general form and then applied to the derivation of geometrically nonlinear bar, and plane stress elements using the Green-Lagrange strain measure. The more complex Generalized CCF (GCCF) is described and applied to the derivation of 2D and 3D Timoshenko beam elements. Several advantages of the CCF, notably the physically clean separation of material and geometric stiffnesses, and its independence with respect to the ultimate choice of shape functions and element degrees of freedom, are noted. Application examples involving very large motions solved with the 3D beam element display the range of applicability of this formulation, which transcends the kinematic limitations commonly attributed to the TL description.

On the spectral stability of time integration algorithms for a class of constrained dynamics problems

Incomplete field formulations have recently been the subject of intense research because of their potential in coupled analysis of independently modeled substructures, adaptive refinement, domain decomposition, and parallel processing. This paper discusses the design and analysis of time-integration algorithms for these formulations and emphasizes the treatment of their inter-subdomain constraint equations. These constraints are shown to introduce a destabilizing effect in the dynamic system that can be analyzed by investigating the behavior of the time-integration algorithm at infinite and zero frequencies. Three different approaches for constructing penalty-free unconditionally stable second-order accurate solution procedures for this class of hybrid formulations are presented, discussed and illustrated with numerical examples. The theoretical results presented in this paper also apply to a large family of nonlinear multibody dynamics formulations. Some of the algorithms outlined herein are important alternatives to the popular technique consisting of transforming differential/algebraic equations into ordinary differential equations via the introduction of a stabilization term that depends on arbitrary constants and that influences the computed so1ution.

Implicit transient finite element structural computations on MIMD systems - FETI vs. direct solvers

A domain decomposition method for implicit schemes that require significantly less storage and is several times faster than factorization algorithms is proposed. The transient domain decomposition method is an extension of the finite element tearing and interconnecting (FETI) method for the solution of static problems. Serial and parallel performance results obtained using the CRAY Y-MP/8 and the iPSC-860/128 systems demonstrate that the FETI method is superior to both serial and parallel direct methods. 15 refs.