Explicit finite element and finite difference methods are used to solve a wide variety of transient problems in industry and academia. Unfortunately, explicit methods are rarely discussed in detail in finite element text books. This paper reviews the basic explicit finite element and finite difference methods that are currently used to solve transient, large deformation problems in solid mechanics. A special emphasis has been placed on documenting methods that have not been previously published in journals.

Computational methods in Lagrangian and Eulerian hydrocodes

This review article concerns a methodology for solving numerically, for engineering purposes, boundary and initial-boundary value problems by a peculiar approach characterized by the following features: the continuous formulation is centered on integral equations based on the combined use of single-layer and double-layer sources, so that the integral operator turns out to be symmetric with respect to a suitable bilinear form. The discretization is performed either on a variational basis or by a Galerkin weighted residual procedure, the interpolation and weight functions being chosen so that the variables in the approximate formulation are generalized variables in Prager’s sense. As main consequences of the above provisions, symmetry is exhibited by matrices with a key role in the algebraized versions; some quadratic forms have a clear energy meaning; variational properties characterize the solutions and other results, invalid in traditional boundary element methods enrich the theory underlying the computational applications. The present survey outlines recent theoretical and computational developments of the title methodology with particular reference to linear elasticity, elastoplasticity, fracture mechanics, time-dependent problems, variational approaches, singular integrals, approximation issues, sensitivity analysis, coupling of boundary and finite elements, and computer implementations. Areas and aspects which at present require further research are identified, and comparative assessments are attempted with respect to traditional boundary integral-elements. This article includes 176 references.

https://hal.archives-ouvertes.fr/hal-00111259/document

Symmetric Galerkin Boundary Element Methods

The accuracies of the computed temperatures of a liquid in a corner region under freezing conditions are compared for various fixed-grid finite element techniques using the analytical solution for this problem as a reference.

In the finite element formulation of the problem different time-stepping schemes are compared: the implicit Euler-backward algorithm combined with an iterative scheme and two three-time-level methods—the Lees algorithm and a Dupont algorithm, which are both applied as non-iterative schemes.

Furthermore, different methods for handling the evolution of latent heat are examined: an approximation method suggested by Lemmon and one suggested by Del Giudice, both using the enthalpy formulation as well as a fictitious heat-flow method presented by Rolph and Bathe.

Results of calculations performed with the consistent heat-capacity matrix are compared with those performed with a lumped heat-capacity matrix.

Comparison of finite element techniques for solidification problems

A three-dimensional (3D) numerical wave tank (NWT) solving fully nonlinear potential flow theory, with a higher-order boundary element method (BEM), is modified to simulate tsunami generation by underwater landslides. New features are added to the NWT to model underwater landslide geometry and motion and specify corresponding boundary conditions in the BEM model. In particular, a new snake absorbing piston boundary condition is implemented to remove reflection from the onshore and offshore boundaries of the NWT. Model results are favorably compared to recent laboratory experiments. Sensitivity analyses of numerical results to the width and length of the discretization are conducted, to determine optimal numerical parameters. The effect of landslide width on tsunami generated is estimated. Results show that the two-dimensional approximation is applicable when the ratio of landslide width over landslide length is greater than 2. Numerical accuracy is examined and found to be excellent in all cases.

Development of a 3D numerical wave tank for modeling tsunami generation by underwater landslides

A “symmetric” boundary element method based on a weighted residual Galerkin approach for elastoplastic analysis is revisited and its computer implementation for two-dimensional homogeneous problems is described.

A galerkin symmetric boundary‐element method in elasticity: Formulation and implementation

This paper discusses the deficiencies of boundary element stiffness matrices, gives an account of the methods proposed to circumvent these defects and proposes a new procedure to obtain symmetric stiffness matrices.

Boundary elements and symmetry

In transient analysis it is generally thought that small time steps can only improve the accuracy, because standard stability theorems limit the maximum time step for a given mesh size. In finite element approximations, however, small time steps may cause stability problems which lead to physically unreasonable results. It is shown that this is due to the violation of a discrete maximum principle. The influence of lumped and consistent mass matrices on a stable discretization of time and space is presented.

On the importance of the discrete maximum principle in transient analysis using finite element methods

What are finite elements?.- What are boundary elements?.- Frames.- Plane problems.- Slabs.- Shells.- Theoretical details.

Structural analysis with finite elements

The boundary element method (BE method) is an integral equation method, or as we could say as well an influence function method. It is based on the fact that in linear problems the boundary values uniquely determine the displacements and stresses inside a structure such as the frame in Fig. 2.1, so that it suffices to discretize the edge with boundary elements only.

What are boundary elements

Finite Elemente : Programme für die Statik in Pascal und C — 2., bearb. und erw. Aufl. (Deutsch) Finite Elemente in der Statik und Dynamik. Link, Michael Finite Elemente in der Baustatik Bücher • Statik mit finiten Elementen • Structural Statik und Einflussfunktionen vom modernen Standpunkt aus, 2. Auflage Das Buch bildet die Grundlage für die Vorlesungsreihe „Finite Elemente“ und „Tragwerksdynamik“, die der Verfasser für Bauingenieurund.

http://link.springer.com/content/pdf/10.1007/978-3-662-58925-0.pdf

C. Katz

Papers

Boundary elements and symmetry

On the importance of the discrete maximum principle in transient analysis using finite element methods

Structural analysis with finite elements

What are boundary elements

Statik mit finiten Elementen