Showing papers in "International Journal for Numerical Methods in Engineering in 1991"

A method of finite element tearing and interconnecting and its parallel solution algorithm

Abstract: A novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented. The spatial domain is partitioned into a set of totally disconnected subdomains, each assigned to an individual processor. Lagrange multipliers are introduced to enforce compatibility at the interface nodes. In the static case, each floating subdomain induces a local singularity that is resolved in two phases. First, the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution. Next, the contributions of these modes are related to the Lagrange multipliers through an orthogonality condition. A parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and Lagrange multipliers, which completes the solution of the problem. When implemented on local memory multiprocessors, this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring. It is also suitable for parallel/vector computers with shared memory. Moreover, unlike parallel direct solvers, it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations.

An alternating digital tree (ADT) algorithm for 3D geometric searching and intersection problems

Abstract: A searching algorithm is presented for determining which members of a set of n points in an N dimensional space lie inside a prescribed space subregion. The algorithm is then extended to handle finite size objects as well as points. In this form it is capable of solving problems such as that of finding the objects from a given set which intersect with a prescribed object. The suitability of the algorithm is demonstrated for the problem of three dimensional unstructured mesh generation using the advancing front method.

Lagrange constraints for transient finite element surface contact

Abstract: A new approach to enforce surface contact conditions in transient non-linear finite element problems is developed in this paper. The method is based on the Lagrange multiplier concept and is compatible with explicit time integration operators. Compatibility with explicit operators is established by referencing Lagrange multipliers one time increment ahead of associated surface contact displacement constraints. However, the method is not purely explicit because a coupled system of equations must be solved to obtain the Lagrange multipliers. An important development herein is the formulation of a highly efficient method to solve the Lagrange multiplier equations. The equation solving strategy is a modified Gauss-Seidel method in which non-linear surface contact force conditions are enforced during iteration. The new surface contact method presented has two significant advantages over the widely accepted penalty function method: surface contact conditions are satisfied more precisely, and the method does not adversely affect the numerical stability of explicit integration. Transient finite element analysis results are presented for problems involving impact and sliding with friction. A brief review of the classical Lagrange multiplier method with implicit integration is also included.

Contact‐impact by the pinball algorithm with penalty and Lagrangian methods

Abstract: Contact-impact algorithms, which are sometimes called slideline algorithms, are a computationally time-consuming part of many explicit simulations of non-linear problems because they involve many branches, so they are not amenable to vectorization, which is essential for speed on supercomputers. The pinball algorithm is a simplified slideline algorithm which is readily vectorized. Its major idea is to embed pinballs in surface elements and to enforce the impenetrability condition only to pinballs. It can be implemented in either a Lagrange multiplier or penalty method. It is shown that, in any Lagrange multiplier method, no iterations are needed to define the contact surface. Examples of solutions and running times are given.

Unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum

Abstract: We show that, for rigid body dynamics, the mid-point rule formulated in body co-ordinates exactly conserves energy and the norm of the angular momentum for incremental force-free motions, but fails to conserve the direction of the angular momentum vector. Further, we show that the mid-point rule formulated in the spatial representation is, in general, physically and geometrically meaningless. An alternative algorithm is developed which exactly preserves energy, and the total spatial angular momentum in incremental force-free motions. The implicit version of this algorithm is unconditionally stable and second order accurate. The explicit version conserves exactly angular momentum in incremental force-free motions. Numerical simulations are presented which illustrate the excellent performance of the proposed procedure, even for incremental rotations over 65 degrees. The procedure is directly applicable to transient dynamic calculations of geometrically exact rods and shells.

A new approach to the development of automatic quadrilateral mesh generation

Abstract: A methodology for the automatic mesh generation of quadrilateral elements is proposed. The methodology is based on the facts that (i) a region can always be subdivided entirely into quadrilaterals if the polygon which forms the boundary of the region has an even number of sides, (ii) a quadrilateral can be formed by two triangles which share a common side. By taking advantage of the techniques developed in an existing automatic triangular mesh generator, quadrilateral meshes can be generated according to the distribution of user specified element size over the domain of interest. In particular, an automatic quadrilateral mesh generator based on the advancing front technique for triangular mesh generation1 is described. Techniques to improve the quality of the generated quadrilateral meshes are introduced. The robustness and versatility of the mesh generator are demonstrated by a variety of examples.

A combined surface integral and finite element solution for a three‐dimensional contact problem

Abstract: A method is described to determine contact stresses and deformation using a combination of the finite element method and a surface integral form of the Bousinesq solution. Numerical examples of contacting hypoid gears are presented.

Adaptivity and mesh generation

Abstract: The objective of achieving economically finite element solutions of specified accuracy is now within reach. The importance of this objective is finally recognized by commercial code originators and efforts directed at automating the process are meeting with success. Three essential ingredients are here necessary. These are: (i) economical and efficient a posteriori error estimating processes; (ii) close prediction of the refinement necessary for a specific accuracy to be achieved so that numerous trial and error solutions can be avoided; (iii) implementation of the predicted refinement. With h and h-p adaptive processes Step (iii) requires powerful mesh generator facilities capable of achieving meshes of specified density. The paper considers all three aspects, introducing some novel concepts in error estimation and discussing in some detail the problem of mesh generation. As such, it introduces the other papers of this special issue, which discuss various aspects in more detail.

Algorithms for draping fabrics on doubly‐curved surfaces

Abstract: Three algorithms for draping biaxially woven fabrics on arbitrarily curved surfaces are presented and compared for numerical accuracy and computational expense. The first one minimizes the elastic energy in each fabric cell, while the two others are based on placing a net of interlocked and inextensible fibres on the surface along geodesic lines. A benchmark shows that the minimum energy technique performs the best and is also the most promising for further optimization in terms of numerical quadrature formulae.

Non‐linear B‐stability and symmetry preserving return mapping algorithms for plasticity and viscoplasticity

Abstract: A class of second order accurate return mapping algorithms is presented which lead to symmetric algorithmic tangent moduli and contain the classical backward-Euler return maps as a particular case. More importantly, it is shown that this class of return maps is contractive relative to the natural norm defined by the complementary Helmholz free energy function (B-stability). Since the equations of classical plasticity and viscoplasticity are shown to be contractive relative to this natural norm, the requirement of B-stability furnishes the appropriate notion of unconditionally stable algorithms for plasticity and viscoplasticity. The analysis that follows depends critically on the assumption of convexity. In particular, the models of plasticity and viscoplasticity considered obey the principle of maximum plastic dissipation. The proposed algorithms obey the discrete counterpart of this classical principle.

Mesh relaxation: A new technique for improving triangulations

Abstract: SUMMARY Given a list of points defining a domain boundary, a three-stage process is often used to triangulate a domain. First, an appropriate distribution of interior points is generated. Next the points are connected to form triangles. And, finally, the connectivity data are used to reposition the interior points using the Laplacian smoothing technique, thereby usually improving the shapes of the triangles. This paper describes a new technique for mesh improvement-adjusting the connection structure during the second stage of this process. The new scheme, which we call mesh relaxation, consists of a procedure for iteratively making the mesh topology more regular by edge swapping. For each interior edge, a relaxation index is computed that depends on the degrees of its end points and adjacent points. Any edge for which this index exceeds a prescribed threshold will be swapped, i.e. replaced by a new edge connecting the adjacent points of the original edge. After all edge swaps are completed, Laplacian smoothing is applied to the mesh. Examples show that, when the mesh point density varies smoothly and due care is taken in the vicinity of the boundary, mesh relaxation can dramatically increase the regularity of the mesh and produce improved triangle shapes.

A refined global‐local finite element analysis method

Abstract: A refined global-local method was proposed to improve the efficiency of finite element analysis The proposed method was based on the regular finite element method in conjunction with three basic step, ie the global analysis, the local analysis and the refined global analysis In the first two steps, a coarse finite element mesh was used to analyse the entire structure to obtain the nodal displacements which were subsequently used as displacement boundary conditions for local regions of interest These local regions with the prescribed boundary conditions were then analysed with refined meshes to obtain more accurate stresses In the third step, a new global displacement distribution based on the results of the previous two steps was assumed for the analysis, from which much improved solutions for both stresses and displacements were produced Numerical examples showed that the proposed method yielded accurate solutions with significant savings in computing time compared with the regular finite element method Further, this method is suitable for parallel computation

Multiplier methods for engineering optimization

Abstract: Multiplier methods used to solve the constrained engineering optimization problem are described. These methods solve the problem by minimizing a sequence of unconstrained problems defined using the cost and constraint functions. The methods, proposed in 1969, have been determined to be quite robust, although not as efficient as other algorithms. They can be more effective for some engineering applications, such as optimum design and control oflarge scale dynamic systems. Since 1969 several modifications and extensions of the methods have been developed. Therefore, it is important to review the theory and computational procedures of these methods so that more efficient and effective ones can be developed for engineering applications. Recent methods that are similar to the multiplier methods are also discussed. These are continuous multiplier update, exact penalty and exponential penalty methods.

Incompressibility without tears—HOW to avoid restrictions of mixed formulation

Abstract: Several time-stepping schemes for incompressibility problems are presented which can be solved directly for steady state or iteratively through the time domain. The difficulty of mixed interpolation is avoided by using these schemes. The schemes are applicable to problems of fluid and solid mechanics.

Representation of geometric stiffening in multibody system simulation

Abstract: Geometric, rotational or dynamic stiffening is a well and long known phenomenon in the analysis of flexible bodies. In multibody dynamics the effect has attracted attention only recently. The objective of this paper is to contribute to the understanding of the modeling of geometric stiffening in multibody system simulation. Today's methods for modeling the effect assume that the (applied) stresses in a flexible system body are zero in its reference configuration, in which it performs large overall motions. The corresponding inertial loads are shown to be balanced by constraint stresses. This can be seen easily when formulating the system equations of motion for nonzero reference stresses. As a result one obtains an efficient alternative to compute the geometric stiffening terms. The method increases the generaltity of flexible body models for multibody system simulation.

The multiple Reciprocity boundary element method in elasticity: A new approach for transforming domain integrals to the boundary

Abstract: This technique operates on domain integrals resulting from different types of body forces such as gravitational and centrifugal forces, as well as loadings due to linear and quadratic temperature distributions.

Solution of Helmholtz equation by Trefftz method

Abstract: The application of the Trefftz method for calculating wave forces on offshore structures is presented. Indirect and direct formulations using complete and non-singular systems of Trefftz functions for the Helmholtz equation are posed in this paper. An effective technique using different interpolation functions for the velocity potential and wave force are suggested to improve the computational accuracy of the wave force. The numerical examples show that the present method is highly efficient and accurate.

A modified Euler method for dynamic analyses

Abstract: A modified version of the well known Euler method for numerical integration is presented, and its application in analyses of the dynamic response of structures is discussed. This modified method was first introduced in 1981, and it has been reported that it was discovered by accident. When first introduced, however, valuable information related to the accuracy, stability and merits of the method were not provided. It is shown that the modified Euler method has features that make it an attractive approach for applications in dynamic analyses of structures. Accuracy and stability analyses are made for the method by considering free-vibrational responses of linear, undamped, single-degree-of-freedom systems. The method is explicit and extremely easy to use. However, it is conditionally stable. Applications of the method are also made which illustrate its usefulness and inherent simplicity. Furthermore, comparisons of the modified Euler method and the Newmark beta method are reported which elucidate the relative merits of these methods.

Automatic conversion of triangular finite element meshes to quadrilateral elements

Abstract: A method is presented for the fully automatic conversion of a general finite element mesh containing triangular elements into a mesh composed of exclusively quadrilateral elements. The initial mesh may be constructed of entirely triangular elements or may consist of a mixture of triangular and quadrilateral elements. The technique used employs heuristic procedures and criteria to selectively combine adjacent triangular elements into quadrilaterals based on preestablished criteria for element quality. Additional procedures are included to eliminate isolated triangles. The methods operates completely without user intervention once the nodal co-ordinates and element connectivity of the original mesh are supplied.

A boundary integral equation method for an inverse problem related to crack detection

Abstract: This paper discusses an application of a boundary integral equation method (BIEM) to an inverse problem of determining the shape and the location of cracks by boundary measurements. Suppose that a given body contains an interior crack, the shape and the location of which are unknown. On the exterior boundary of this body one carries out measurements which are interpreted mathematically as prescribing Dirichlet data and measuring the corresponding Neumann data, or vice versa, for a field governed by Laplace's equation. The inverse problem considered here attempts to determine the geometry of the crack from these experimental data. We propose to solve this problem by minimizing the error of a certain boundary integral equation (BIE). The process of this minimization, however, is shown to require solutions of certain are proposed. Several 2D and 3D numerical examples are given in order to test the performance of the present method.

An investigation of a finite rotation four node assumed strain shell element

Abstract: The paper presents a shell formulation based on the ‘degenerated solid approach’. The theory employs covariant strains and performs explicit integration through the shell thickness. The rigid body motion is exactly represented. The consistent tangent stiffness matrix is evaluated for the four node quadrilateral. It is shown, in the final part, that this type of element, which distinguishes itself by a very simple and easily understandable theory, gives good answers for linear as well as non-linear applications.

Delaunay versus max‐min solid angle triangulations for three‐dimensional mesh generation

Abstract: The Delaunay triangulation has been used in several methods for generating finite element tetrahedral meshes in three-dimensional polyhedral regions. Other types of three-dimensional triangulations are possible, such as a triangulation satisfying a local max-min solid angle criterion. In this paper, we present experimental results to show that max-min solid angle triangulations are better than Delaunay triangulations for finite element tetrahedral meshes, since the former type of triangulations contains tetrahedra of better shape than the latter type. We also describe how mesh points are generated and triangulated in our tetrahedral mesh generation method.

The Lanczos algorithm applied to unsymmetric generalized eigenvalue problem

Abstract: Application of the two-sided Lanczos recursion to the unsymmetric generalized eigenvalue problem is presented. The system matrices are real and unsymmetric. Therefore, the recursions are performed in real arithmetic and complex arithmetic is employed in the QR algorithm used to extract the eigenvalues of the transformed tridiagonal matrix. The biorthonormal transformation of the unsymmetric generalized eigenvalue problem is considered in detail with appropriate proofs presented in Appendices. Issues relating to the computer implementation of the unsymmetric generalized eigenvalue problem are discussed. The example problems solved demonstrate the working of the algorithm in extracting the complex and/or real eignevalues of an unsymmetric system of matrices. Also, the algorithm is applied to extract a few of the eigenvalues of a large fluid-structure interaction problem, and the results are compared with the eigenfrequencies extracted by an unsymmetric subspace iteration procedure presented in the literature.

Efficient boundary element analysis of sharp notched plates

Abstract: The present paper further develops the boundary element singularity subtraction technique, to provide an efficient and accurate method of analysing the general mixed-mode deformation of two-dimensional linear elastic structures containing sharp notches. The elastic field around sharp notches is singular. Because of the convergence difficulties that arise in numerical modelling of elastostatic problems with singular fields, these singularities are subtracted out of the original elastic field, using the first term of the Williams series expansion. This regularization procedure introduces the stress intensity factors as additional unknowns in the problem; hence extra conditions are required to obtain a solution. Extra conditions are defined such that the local solution in the neighbourhood of the notch tip is identical to the Williams solution; the procedure can take into account any number of terms of the series expansion. The standard boundary element method is modified to handle additional unknowns and extra boundary conditions. Analysis of plates with symmetry boundary conditions is shown to be straightforward, with the modified boundary element method. In the case of non-symmetrical plates, the singular tip-tractions are not primary boundary element unknowns. The boundary element method must be further modified to introduce the boundary integral stress equations of an internal point, approaching the notch-tip, as primary unknowns in the formulation. The accuracy and efficiency of the method is demonstrated with some benchmark tests of mixed-mode problems. New results are presented for the mixed-mode analysis of a non-symmetrical configuration of a single edge notched plate.

Modelling of superelements in mechanism analysis

Abstract: The paper deals with substructuring for dynamic analysis of flexible multibody systems. Three different techniques based on component synthesis are discussed, corresponding respectively to fully consistent mass discretization, lumped mass discretization and corotational approximation of inertia forces. To simplify the computer implementation, only the lumped mass and corotational approximations have been considered in detail and programmed. Both approaches are validated on simple examples of rotating beams for which a full elastic model is available using a fully non-linear beam element. The computational efficiency of the corotational inertia approach is also demonstrated on the deployment of a large flexible satellite antenna.

Exact block diagonalization of large eigenvalue problems for structures with symmetry

Abstract: We consider large eigenvalue problems for skeletal structures with symmetry. We present an algorithm, based upon a novel combination of group-theoretic ideas and substructuring techniques, that block-diagonalizes such systems exactly and efficiently. The procedure requires only the structural matrices of a repeating substructure, together with the symmetry modes, which are obtained from symmetry considerations alone. We first present a simple paradigmatic example and then follow with several non-trivial applications involving large lattice structures.

Application of the boundary element method to transient heat conduction

Abstract: An advanced boundary element method (BEM) is presented for the transient heat conduction analysis of engineering components. The numerical implementation necessarily includes higher-order conforming elements, self-adaptive integration and a multiregion capability. Planar, three-dimensional and axisymmetric analyses are all addressed with a consistent time-domain convolution approach, which completely eliminates the need for volume discretization for most practical analyses. The resulting general purpose algorithm establishes BEM as an attractive alternative to the more familiar finite difference and finite element methods for this class of problems. Several detailed numerical examples are included to emphasize the accuracy, stability and generality of the present BEM. Furthermore, a new efficient treatment is introduced for bodies with embedded holes. This development provides a powerful analytical tool for transient solutions of components, such as casting moulds and turbine blades, which are cumbersome to model when employing the conventional domain-based methods.

Solid elements with rotational degrees of freedom: Part II—tetrahedron elements

Abstract: This is the first of a two part paper on three-dimensional finite elements with rotational degrees of freedom (DOF). Part I introduces an 8-node solid hexahedron element having three translational and three rotational DOF per node. The corner rotations are introduced by transformation of the midside translational DOF of a 20-node hexahedron element. The new element produces a much smaller effective band width of the global system equations than does the 20-node hexahedron element having midside nodes. A small penalty stiffness is introduced to augment the usual element stiffness so that no spurious zero energy modes are present. The new element passes the patch test and demonstrates greatly improved performance over elements of identical shape but having only translational DOF at the corner nodes.