Numerical Methods : Scipedia

Shape variable definition with C0 , C1 and C2 continuity functions
G. Chiandussi, G. Bugeda, E. Oñate

Comput. Methods Appl. Mech. Engrg., (2000). Vol. 188, pp. 727-742

Abstract
The present paper proposes a new technique for the definition of the shape design variables in 2D and 3D optimisation problems. It can be applied to the discrete model of the analysed structure or to the original geometry without any previous knowledge of the analytical expression of the CAD defining surfaces. The proposed technique allows the surface continuity to be preserved during the geometry modification process to be defined a priori. This capability allows for the definition of shape variables suitable for every kind of discipline involved in the optimisation process (structural analysis, fluid-dynamic analysis, crash analysis, aerodynamic analysis, etc.).
Abstract
The present paper proposes a new technique for the definition of the shape design variables in 2D and 3D optimisation problems. It can be applied to the discrete model of [...]
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Error estimation including pollution assessment for nonlinear finite element analysis
A. Huerta, P. Diez

Comput. Methods Appl. Mech. Engrg., (2000). Vol. 181 (1-3), pp. 21-41

Abstract
A residual type error estimator for nonlinear finite element analysis is introduced. This error estimator solves local problems avoiding both the computation of the flux jumps and the associated flux splitting procedure. Pollution errors are taken into account by a feedback strategy, that is, an error estimate based on local computations is used as the input of the pollution analysis. This estimator is used in the frame of an adaptive procedure. Numerical examples show that the estimator is able to drive adaptive procedures leading to likely good solutions. Moreover, one of the examples demonstrates that adaptive procedures are essential for complex highly nonlinear mechanical problems because they may discover secondary collapse mechanisms.
Abstract
A residual type error estimator for nonlinear finite element analysis is introduced. This error estimator solves local problems avoiding both the computation of the flux jumps [...]
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A simple method for automatic update of finite element meshes
G. Chiandussi, G. Bugeda, E. Oñate

Commun. Numer. Meth. Engng (2000). Vol. 16 (1), pp. 1-19

Abstract
A simple method to automatically update the finite element mesh of the analysis domain is proposed. The method considers the mesh as a fictitious elastic body subjected to prescribed displacements at selected boundary points. The mechanical properties of each mesh element are appropriately selected in order to minimize the deformation and the distortion of the mesh elements. Different selection strategies have been used and compared in their application to simple examples. The method avoids the use of remeshing in the solution of shape optimization problems and reduces the number of remeshing steps in the solution of coupled fluid–structure interaction problems
Abstract
A simple method to automatically update the finite element mesh of the analysis domain is proposed. The method considers the mesh as a fictitious elastic body subjected to [...]
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Adaptive finite element strategies based on error assessment
A. Huerta, A. Rodríguez-Ferran, P. Diez, J. Sarrate

Int. J. Numer. Meth. Engng. (1999). Vol. 46 (10), pp. 1803-1818

Abstract
Two main ingredients are needed for adaptive finite element computations. First, the error of a given solution must be assessed, by means of either error estimators or error indicators. After that, a new spatial discretization must be defined via h, p or r-adaptivity. In principle, any of the approaches for error assessment may be combined with any of the procedures for adapting the discretization. However, some combinations are clearly preferable. The advantages and limitations of the various alternatives are discussed. The most adequate strategies are illustrated by means of several applications in solid mechanics.
Abstract
Two main ingredients are needed for adaptive finite element computations. First, the error of a given solution must be assessed, by means of either error estimators or error [...]
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A unified approach to remeshing strategies for finite element h-adaptivity
P. Diez, A. Huerta

Comput. Methods Appl. Mech. Engrg., (1999). Vol. 176 (1-4), pp. 215-229

Abstract
In h-adaptivity, a remeshing strategy is needed to compute the distribution of required element size using the estimated error distribution. Several authors have introduced different remeshing strategies yielding very different results. In this work these methods are included in a unified framework, emphasizing the role of the underlying hypotheses. Moreover, an objective tool to evaluate the accuracy of the resulting finite element solution is presented. Thus, a new remeshing strategy is introduced to optimize the accuracy of the adapted solutions. The different remeshing strategies are compared with well-known numerical examples.
Abstract
In h-adaptivity, a remeshing strategy is needed to compute the distribution of required element size using the estimated error distribution. Several authors have introduced [...]
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A posteriori error estimation for standard finite element analysis
P. Díez, J. Egozcue, A. Huerta

Comput. Methods Appl. Mech. Engrg., (1998). Vol. 163 (1-4), pp. 141-157

Abstract
A new residual type estimator based on projections of the error on subspaces of locally-supported functions is presented. The estimator is defined by a standard element-by-element refinement. First, an approximation of the energy norm of the error is obtained solving local problems with homogeneous Dirichlet boundary conditions. A later enrichment of the estimation is performed by adding the contributions of projections on a new family of subspaces. This estimate is a lower bound of the measure of the actual error. The estimator does not need to approximate local boundary conditions for the error equation. Therefore, computation of flux jumps is not necessary. Moreover, the estimator can be applied in mixed meshes containing elements of different shapes and its implementation in a standard finite element code is straightforward. The presented results show the effectiveness of the estimator approximating both the distribution and the global measure of the error, as well as its usefulness in adaptive procedures.
Abstract
A new residual type estimator based on projections of the error on subspaces of locally-supported functions is presented. The estimator is defined by a standard element-by-element [...]
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Shock capturing viscosities for the general fluid mechanics algorithm
P. Nithiarasu, O. Zienkiewicz, B. Sai, K. Morgan, R. Codina, M. Vázquez

Int. J. Numer. Meth. Fluids (1998). Vol. 28 (9), pp. 1325-1353

Abstract
The performance of different shock capturing viscosities has been examined using our general fluid mechanics algorithm. Four different schemes have been tested, both for viscous and inviscid compressible flow problems. Results show that the methods based on the second gradient of pressure give better performance in all situations. For instance, the method constructed from the nodal pressure values and consistent and lumped mass matrices is an excellent choice for inviscid problems. The method based on L2 projection is better than any other method in viscous flow computations. The residual based anisotropic method gives excellent performance in the supersonic range and gives better results in the hypersonic regime if a small amount of residual smoothing is used
Abstract
The performance of different shock capturing viscosities has been examined using our general fluid mechanics algorithm. Four different schemes have been tested, both for viscous [...]
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An advancing front point generation technique
R. Löhner, E. Oñate

Commun. Numer. Meth. Engng (1998). Vol. 14 (12), pp. 1097-1108

Abstract
An algorithm to construct boundary‐conforming, isotropic clouds of points with variable density in space is described. The input required consists of a specified mean point distance and an initial triangulation of the surface. Borrowing a key concept from advancing front grid generators, one point at a time is removed and, if possible, surrounded by admissible new points. This operation is repeated until no active points are left. Timings show that the scheme is about an order of magnitude faster than volume grid generators based on the advancing front technique, making it possible to generate large (>106) yet optimal clouds of points in a matter of minutes on a workstation. Several examples are included that demonstrate the capabilities of the technique.
Abstract
An algorithm to construct boundary‐conforming, isotropic clouds of points with variable density in space is described. The input required consists of a specified mean point [...]
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On the computational efficiency and implementation of block-iterative algorithms for nonlinear coupled Problems
M. Cervera, R. Codina, M. Galindo

Engineering Computations (1996). Vol. 13 (6), pp. 4-30

Abstract
Outlines a general methodology for the solution of the system of algebraic equations arising from the discretization of the field equations governing coupled problems. Considers that this discrete problem is obtained from the finite element discretization in space and the finite difference discretization in time. Aims to preserve software modularity, to be able to use existing single field codes to solve more complex problems, and to exploit computer resources optimally, emulating parallel processing. To this end, deals with two well-known coupled problems of computational mechanics – the fluid-structure interaction problem and thermally-driven flows of incompressible fluids. Demonstrates the possibility of coupling the block-iterative loop with the nonlinearity of the problems through numerical experiments which suggest that even a mild nonlinearity drives the convergence rate of the complete iterative scheme, at least for the two problems considered here. Discusses the implementation of this alternative to the direct coupled solution, stating advantages and disadvantages. Explains also the need for online synchronized communication between the different codes used as is the description of the master code which will control the overall algorithm.
Abstract
Outlines a general methodology for the solution of the system of algebraic equations arising from the discretization of the field equations governing coupled problems. Considers [...]
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Finite volumes and finite elements: Two ‘good friends’
E. Oñate, S. Idelsohn

Int. J. Numer. Meth. Engng. (1994). Vol. 37 (19), pp. 3323-3341

Abstract
In this paper a comparison between the finite element and the finite volume methods is presented in the context of elliptic, convective–diffusion and fluid flow problems. The paper shows that both procedures share a number of features, like mesh discretization and approximation. Moreover, it is shown that in many cases both techniques are completely equivalent.
Abstract
In this paper a comparison between the finite element and the finite volume methods is presented in the context of elliptic, convective–diffusion and fluid flow problems. [...]
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A study of mesh optimality criteria in adaptive finite element analysis
E. Oñate, G. Bugeda

Engineering Computations (1993). Vol. 10 (4), pp. 307-321

Abstract
The concepts of solution error and optimal mesh in adaptive finite element analysis are revisited. It is shown that the correct evaluation of the convergence rate of the error norms involved in the error measure and the optimal mesh criteria chosen are essential to avoid oscillations in the refinement process. Two mesh optimality criteria based on: (a) the equal distribution of global error, and (b) the specific error over the elements are studied and compared in detail through some examples of application.
Abstract
The concepts of solution error and optimal mesh in adaptive finite element analysis are revisited. It is shown that the correct evaluation of the convergence rate of the error [...]
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Partitioned analysis of coupled mechanical systems
C. Felippa, K. Park, C. Farhat

Computer Methods in Applied Mechanics and Engineering (2001). Vol. 190 (24-25), pp. 3247-3270

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.
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 [...]
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An alphanumeric finite element mesh plotter
C. Felippa

International Journal for Numerical Methods in Engineering (1972). Vol. 5 (2), pp. 217-236

Abstract
A FORTRAN subroutine name MESHPL is described. This routine generates an alphanumeric image of an arbitrary two-dimensional finite element mesh on the line printer. This plotting technique can be used for fast on-line display of nodal points, elements, boundary conditions and loads during the input data preprocessing stage of a finite element analysis.
Abstract
A FORTRAN subroutine name MESHPL is described. This routine generates an alphanumeric image of an arbitrary two-dimensional finite element mesh on the line printer. This plotting [...]
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Solution of linear equations with skyline-stored symmetric matrix
C. Felippa

Computers and Structures (1975). Vol. 5 (1), pp. 13-29

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.
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 [...]
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Optimization of finite element grids by direct energy search
C. Felippa

Applied Mathematical Modelling (1976). Vol. 1 (2), pp. 93-96

Abstract
One of the possible approaches to the reduction of computational costs in finite element analysis is the selection of ‘optimal grids’, which produce the ‘best’ answers, in the sense of minimizing a discretization error measure, for a fixed level of computational effort. The grid optimization problem is studied in the case of grids of similar topology having a fixed number of degrees of freedom per node. A general formulation based on weighted-residual error measures is specialized to field problems associated with a positive-definite energy functional, the minimization of which, with respect to variable node locations, is adopted as a grid optimality criterion. The problem is then embedded in the framework of the general nonlinear programming problem, and desirable computational features of candidate search algorithms are described.
Abstract
One of the possible approaches to the reduction of computational costs in finite element analysis is the selection of ‘optimal grids’, which produce the [...]
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Error analysis of penalty function techniques for constraint definition in linear algebraic systems
C. Felippa

International Journal for Numerical Methods in Engineering (1977). Vol. 11 (4), pp. 709-728

Abstract
The penalty function approach has been recently formalized as a general technique for adjoining constraint conditions to algebraic equation systems resulting from variational discretization of boundary value problems by finite difference or finite element techniques. This paper studies the numerical behaviour of the penalty function method for the special case of individual equation constraints imposed on a symmetric system of linear algebraic equations. Constraint representation and computational roundoff error components are distinguished and asymptotically characterized in terms of the penalty function weight coefficients. On the basis of this study, practical rules for the automatic assignment of values to those coefficients within the linear equation solver are proposed. Numerical problems encountered in the case of more general constraints are briefly discussed, and procedures for circumventing such difficulties are suggested.
Abstract
The penalty function approach has been recently formalized as a general technique for adjoining constraint conditions to algebraic equation systems resulting from variational [...]
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Numerical experiments in finite element grid optimization by direct energy search
C. Felippa

Applied Mathematical Modelling (1977). Vol. 1 (5), pp. 239-244

Abstract
A series of numerical experiments is conducted to assess the feasibility and practical value of finite element grid optimization based on direct minimization of the total potential energy of the discrete model with respect to the node locations. An implementation relying upon non-linear programming techniques is found to be numerically reliable and to lead to improved grids in accord with engineering intuition. This rigorous approach is hampered, however, by the excessive computational effort required by the energy minimization process. A combination of related techniques is therefore proposed to make dynamic node distribution a useful tool within the framework of large-scale finite element analysis. The combined strategy involves use of substructuring methods, application of a local energy-balancing optimality criterion for fast node distribution, and automatic refinement of previously-improved coarse grids.
Abstract
A series of numerical experiments is conducted to assess the feasibility and practical value of finite element grid optimization based on direct minimization of the total [...]
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Iterative solutions for improving penalty function solutions of algebraic systems
C. Felippa

International Journal for Numerical Methods in Engineering (1978). Vol. 12 (5), pp. 821-836

Abstract
The standard implementation of the penalty function approach for the treatment of general constraint conditions in discrete systems of equations often leads to computational difficulties as the penalty weights are increased to meet constraint satisfaction tolerances. A family of iterative procedures that converges to the constrained solution for fixed weights is presented. For a discrete mechanical system, these procedures can be physically interpreted as an equilibrium iteration resulting from the appearance of corrective force patterns at the nodes of ‘constraint members’ of constant stiffness. Three forms of the iteration algorithm are studied in detail. Convergence conditions are established and the computational error propagation behaviour of the three forms is analysed. The conclusions are verified by numerical experiments on a model problem. Finally, practical guidelines concerning the implementation of the corrective process in large-scale finite element codes are offered.
Abstract
The standard implementation of the penalty function approach for the treatment of general constraint conditions in discrete systems of equations often leads to computational [...]
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A contour-line printplotter for finite element work
C. Felippa

Advances in Engineering Software (1979). Vol. 1 (2), pp. 51-60

Abstract
A Fortran subroutine for producing countour line printplots of functions defined at the nodes of an arbitrary two-dimensional finite element mesh is presented. This display technique is useful for quick inspection of analysis results and as guide for downstream use of more sophisticated graphics equipment.
Abstract
A Fortran subroutine for producing countour line printplots of functions defined at the nodes of an arbitrary two-dimensional finite element mesh is presented. This display [...]
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Staggered transient analysis procedures for coupled mechanical systems: Formulation
C. Felippa, K. Park

Computer Methods in Applied Mechanics and Engineering (1980). Vol. 24 (1), pp. 61-111

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.
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 [...]
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Shape variable definition with C0 , C1 and C2 continuity functions

Error estimation including pollution assessment for nonlinear finite element analysis

A simple method for automatic update of finite element meshes

Adaptive finite element strategies based on error assessment

A unified approach to remeshing strategies for finite element h-adaptivity

A posteriori error estimation for standard finite element analysis

Shock capturing viscosities for the general fluid mechanics algorithm

An advancing front point generation technique

On the computational efficiency and implementation of block-iterative algorithms for nonlinear coupled Problems

Finite volumes and finite elements: Two ‘good friends’

A study of mesh optimality criteria in adaptive finite element analysis

Partitioned analysis of coupled mechanical systems

An alphanumeric finite element mesh plotter

Solution of linear equations with skyline-stored symmetric matrix

Optimization of finite element grids by direct energy search

Error analysis of penalty function techniques for constraint definition in linear algebraic systems

Numerical experiments in finite element grid optimization by direct energy search

Iterative solutions for improving penalty function solutions of algebraic systems

A contour-line printplotter for finite element work

Staggered transient analysis procedures for coupled mechanical systems: Formulation

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