Carlos A. Felippa

Partitioned analysis of coupled mechanical systems

Abstract This is a tutorial article that reviews the use of partitioned analysis procedures for the analysis of coupled dynamical systems. Attention is focused on the computational simulation of systems in which a structure is a major component. Important applications in that class are provided by thermomechanics, fluid–structure interaction and control–structure interaction. In the partitioned solution approach, systems are spatially decomposed into partitions. This decomposition is driven by physical or computational considerations. The solution is separately advanced in time over each partition. Interaction effects are accounted for by transmission and synchronization of coupled state variables. Recent developments in the use of this approach for multilevel decomposition aimed at massively parallel computation are discussed.

A triangular membrane element with rotational degrees of freedom

Abstract A new plane-stress triangular element is derived using the free formulation of Bergan and Nygard. The triangle possesses nine degrees of freedom: six corner translations and three corner normal rotations. The element is coordinate-invariant and passes the patch test for any geometry. Two free parameters in the formulation may be adjusted to optimize the behavior for in-plane bending patterns. With the recommended parameter choices the element performance is significantly better than that of the constant-strain triangle. Because of the presence of the rotational freedoms, this new element appears especially suitable as membrane component of a flat triangular element for modelling general shell structures.

Staggered transient analysis procedures for coupled mechanical systems: Formulation

Abstract Coupled-field dynamic problems in mechanics have been traditionally solved by treating the entire system as one computational entity. More recently, increasing attention has been directed to an alternative approach; partition the governing equations into subsystems, which are treated by subsystem analyzers. The selection of the subsystems may be based on weak-coupling considerations, widely different time response characteristics, isolation of nonlinear effects, or more pragmatic reasons such as the availability of analyzer software. In a staggered solution procedure the solution state of the coupled system is advanced by sequentially executing the subsystem analyzers. Subsystem coupling terms are accounted for by temporal extrapolation techniques. This paper focuses on the formulation and computer implementation of staggered solution procedures for two-field problems governed by semidiscrete second-order coupled differential equations. Such equations find application in the modeling of structure-fluid, structure-soil and structure-structure interaction. Following an introductory description of candidate problems and general solution strategies, direct time integration methods are formulated and applied to the coupled system. Staggered solution procedures are constructed through two alternative approaches which are based upon partitioning at the difference and differential equation level, respectively. Characteristic equations that govern the stability of the resulting implementations are derived, and the selection of stable extrapolators discussed. Finally, possible extensions of staggered solution procedures to coupled-field static and eigenvalue problems are suggested.

A study of optimal membrane triangles with drilling freedoms

Abstract This article compares derivation methods for constructing optimal membrane triangles with corner drilling freedoms. The term “optimal” is used in the sense of exact inplane pure-bending response of rectangular mesh units of arbitrary aspect ratio. Following a comparative summary of element formulation approaches, the construction of an optimal three-node triangle using the ANDES formulation is presented. The construction is based upon techniques developed by 1991 in student term projects, but taking advantage of the more general framework of templates developed since. The optimal element that fits the ANDES template is shown to be unique if energy orthogonality constraints are enforced. Two other formulations are examined and compared with the optimal model. Retrofitting the conventional linear strain triangle element by midpoint-migrating and congruential transformations is shown unable to produce an optimal element, while rank deficiency is inevitable. Use of the quadratic strain field of the 1988 Allman triangle, or linear filtered versions thereof, is also unable to reproduce the optimal element. Moreover these elements exhibit serious aspect ratio lock. These predictions are verified on benchmark examples.

A variational principle for the formulation of partitioned structural systems

A continuum-based variational principle is presented for the formulation of the discrete governing equations of partitioned structural systems. This application includes coupled substructures as well as subdomains obtained by mesh decomposition. The present variational principle is derived by a series of modifications of a hybrid functional originally proposed by Atluri for finite element development. The interface is treated by a displacement frame and a localized version of the method of Lagrange multipliers. Interior displacements are decomposed into rigid-body and deformational components to handle floating subdomains. Both static and dynamic versions are considered. An important application of the present principle is the treatment of nonmatching meshes that arise from various sources such as separate discretization of substructures, independent mesh refinement, and global–local analysis. The present principle is compared with that of a globalized version of the multiplier method. Copyright

Partitioned Transient Analysis Procedures for Coupled-Field Problems: Accuracy Analysis

Abstract : A general partitioned transient analysis procedure is proposed, which is amenable to a unified stability analysis technique. The procedure embodies two existing inplicit-explicit procedures and one existing implicit-implicit procedure. A new implicit-explicit procedure is discovered, as a special case of the general procedure, that allows degree-by-degree implicit or explicit selections of the solution vector and can be implemented within the framework of the implicit integration packages. A new element-by-element implicit-implicit procedure is also presented which satisfies program modularity requirements and enables the use of single-field implicit integration packages to solve coupled-field problems. (Author)

Solution of linear equations with skyline-stored symmetric matrix

Abstract Fortran IV subroutines for the in-core solution of linear algebraic systems with a sparse, symmetric, skyline-stored coefficient matrix are presented. Such systems arise in a variety of applications, notably the numerical discretization of conservative physical systems by finite differences or finite element techniques. The routines can be used for processing constrained systems without need for prearranging equations. The application to ‘superelement’ condensation of large-scale systems is discussed.

AN ALGEBRAICALLY PARTITIONED FETI METHOD FOR PARALLEL STRUCTURAL ANALYSIS : ALGORITHM DESCRIPTION

An algebraically partitioned FETI method for the solution of structural engineering problems on parallel computers is presented. The present algorithm consists of three attributes: an explicit generation of the orthogonal null-space matrix associated with the interface nodal forces, the floating subdomain rigid-body modes computed from the subdomain static equilibrium equation of the classical force method and the identification of redundant interface force constraint operator that emanates when the interface force computations are localized. Comparisons of the present method with the previously developed differentially partitioned FETI method are offered in terms of the saddle-point formulations at the end of the paper. A companion paper reports implementation details and numerical performance of the proposed algorithm.

A Variational Framework for Solution Method Developments in Structural Mechanics

We present a variational framework for the development of partitioned solution algorithms in structural mechanics. This framework is obtained by decomposing the discrete virtual work of an assembled structure into that of partitioned substructures in terms of partitioned substructural deformations, substructural rigid-body displacements and interface forces on substructural partition boundaries. New aspects of the formulation are: the explicit use of substructural rigid-body mode amplitudes as independent variables and direct construction of rank-sufficient interface compatibility conditions. The resulting discrete variational functional is shown to be variationally complete, thus yielding a full-rank solution matrix. Four specializations of the present framework are discussed. Two of them have been successfully applied to parallel solution methods and to system identification. The potential of the two untested specializations is briefly discussed.

The first ANDES elements: 9-dof plate bending triangles

Abstract New elements are derived to validate and assess the assumed natural deviatoric strain (ANDES) formulation. This is a brand new variant of the assumed natural strain (ANS) formulation of finite elements, which has recently attracted attention as an effective method for constructing high-performance elements for linear and nonlinear analysis. The ANDES formulation is based on an extended parametrized variational principle developed in recent publications. The key concept is that only the deviatoric part of the strains is assumed over the element, whereas the mean strain part is discarded in favor of a constant stress assumption. Unlike conventional ANS elements, ANDES elements satisfy the individual element test (a stringent form of the patch test) a priori while retaining the favorable distortion-insensitivity properties of ANS elements. The first application of this new formulation is the development of several Kirchhoff plate bending triangular elements with the standard nine degrees of freedom. Linear curvature variations are sampled along the three sides with the corners as ‘gage reading’ points. These samples values are interpolated over the triangle using three schemes. Two schemes merge back to conventional ANS elements, one being identical to the discrete Kirchhoff triangle (DKT), whereas the third one products two new ANDES elements. Numerical experiments indicate that one of the ANDES element is relatively insensitive to distortion compared to previously derived high-performance plate-bending elements, while retaining accuracy for nondistorted elements.