On computational procedures for the force method

It is known that the matrix force method has certain advantages over the displacement method for a class of structural problems. It is also known that the force method, when carried out by the conventional Gauss-Jordan procedure, tends to fill in the problem data, making the method unattractive for large size, sparse problems. This poor fill-in property, however, is not necessarily inherent to the method, and the sparsity may be maintained if one uses what we call the Turn-Back LU Procedure. The purpose of this paper is two-fold. First, it is shown that there exist some close relationships between the force method and the least squares problem, and that many existing algebraic procedures to perform the force method can be regarded as applications/extensions of certain well-known matrix factorization schemes for the least squares problem. Secondly, it is demonstrated that these algebraic procedures for the force method can be unified form the matrix factorization viewpoint. Included in this unification is the Turn-Back LU Procedure, which was originally proposed by Topçu in his thesis.⁸ It is explained why this procedure tends to produce sparse and banded ‘self-stress’ and flexibility matrices with small band width. Some computational results are presented to demonstrate the superiority of the Turn-Back LU Procedure over the other schemes considered in this paper.     

A new four-node quadrilateral plate bending element for highly sparse and banded flexibility matrices

In this paper, an efficient method is developed for the formation of null bases of four-node quadrilateral plate bending finite element models, corresponding to highly sparse and banded flexibility matrices. This is achieved by introducing a new four-node quadrilateral plate bending element, and using special graphs associated with the finite element models. The results are compared to those of the previously developed graph theoretical and algebraic force methods, and also the displacement approach.     

A new implementation of the force method and the optimum design of large trusses

An implementation of the force method is proposed in which the forces and the displacements are simultaneously obtained by the solution of a sparse symmetric indefinite system. The matrix of coefficients is formed by just the concatenation of the element flexibility and equilibrium matrices. No computational procedure is required to generate the compatibility conditions (or the self-stress matrix) and no partitioning of the force vector is made into a basic set and a redundant set, unlike the conventional force method. A slightly modified sparse unsymmetric system can be written in which the stresses and the displacements are the unknowns. This is used as constraints in the formulation of the minimum weight design problem for large structures under static loading conditions. A sparse generalized reduced gradient package is used as the optimizer. A class of test problems involving large truss structures is solved. The results indicate that the present implementation of the force method is better than the displacement method for the optimum design of large structures.     

Efficient finite element analysis of models comprised of higher order triangular elements

This paper introduces an efficient method for the finite element analysis of models comprised of higher order triangular elements. The presented method is based on the force method and benefits graph theoretical transformations. For this purpose, minimal subgraphs of predefined special patterns are selected. Self-equilibrating systems are then constructed on these subgraphs leading to sparse and banded null basis. Finally, well-structured flexibility matrices are formed for efficient finite element analysis.     

Direct Imaging of the Induced‐Fit Effect in Molecular Self‐Assembly

Molecular recognition is a crucial driving force for molecular self‐assembly. In many cases molecules arrange in the lowest energy configuration following a lock‐and‐key principle. When molecular flexibility comes into play, the induced‐fit effect may govern the self‐assembly. Here, the self‐assembly of dicyanovinyl‐hexathiophene (DCV6T) molecules, a prototype specie for highly efficient organic solar cells, on Au(111) by using low‐temperature scanning tunneling microscopy and atomic force microscopy is investigated. DCV6T molecules assemble on the surface forming either islands or chains. In the islands the molecules are straight—the lowest energy configuration in gas phase—and expose the dicyano moieties to form hydrogen bonds with neighbor molecules. In contrast, the structure of DCV6T molecules in the chain assemblies deviates significantly from their gas‐phase analogues. The seemingly energetically unfavorable bent geometry is enforced by hydrogen‐bonding intermolecular interactions. Density functional theory calculations of molecular dimers quantitatively demonstrate that the deformation of individual molecules optimizes the intermolecular bonding structure. The intermolecular bonding energy thus drives the chain structure formation, which is an expression of the induced‐fit effect.     

Parallel implementation of spectral element method for Lamb wave propagation modeling

The proposed spectral element method implementation is based on sparse matrix storage of local shape function derivatives calculated at Gauss–Lobatto–Legendre points. The algorithm utilizes two basic operations: multiplication of sparse matrix by vector and element‐by‐element vectors multiplication. Compute‐intensive operations are performed for a part of equation of motion derived at the degree of freedom level of 3D isoparametric spectral elements. The assembly is performed at the force vector in such a way that atomic operations are minimized. This is achieved by a new mesh coloring technique The proposed parallel implementation of spectral element method on GPU is applied for the first time for Lamb wave simulations. It has been found that computation on multicore GPU is up to 14 times faster than on single CPU. Copyright © 2015 John Wiley & Sons, Ltd.     

Application of the two-dimensional Hermitian finite difference method to linear shear deformation theory of plates and arbitrarily curved shells

Summary Starting from the linear, partial differential equations of thin shell theory including the effect of transverse shear deformation a matrix formulation is presented in which all kinematic and dynamic variables appear as differential quotients of first order.The transformation of the local field equations into an algebraic form by appropriate two-dimensional finite-difference operators leads to an unsymmetrical banded algebraic equation system from which all variables requested are directly evaluated.The constitutive equations take into consideration a symmetrically layered cross section consisting of tangentially isotropic and transversally orthotropic material. Because of the kinematic assumptions of a first approximation shell theory the deformation of the cross section can be seized in an integral sense only.Since the basic equations are formulated in tensor notation with the procedure presented shells of arbitrary curvature may be analyzed, of which surface is given as an analytic continuously differentiable function. The algorithm pointed out shows good convergence attributes. Its efficiency is demonstrated by two examples of which analytical solutions are known.     

Shifted FSAI preconditioners for the efficient parallel solution of non‐linear groundwater flow models

The present paper investigates the performance of a shifted factorized sparse approximate inverse as a parallel preconditioner for the iterative solution to the linear systems arising in the finite element discretization of non‐linear groundwater flow models. The shift strategy is based on an inexpensive preconditioner update exploiting the structure of the coefficient matrix. The proposed algorithm is experimented with in the parallel simulation of a large‐scale real multi‐aquifer system characterized by a stochastic distribution of the hydraulic conductivity. The numerical results show that the shifted factorized sparse approximate inverse algorithm may yield an overall computational gain up to 300% with respect to the non‐shifted scheme with an excellent parallel efficiency. Copyright © 2011 John Wiley & Sons, Ltd.     

Stress‐adapted numerical form finding of pre‐stressed surfaces by the updated reference strategy

In this paper we present the updated reference strategy for numerical form finding of pre‐stressed membranes, which is based on standard finite element discretization. The singularities of the inverse problem are regularized by a homotopy mapping. A projection scheme is proposed where anisotropic pre‐stress is defined with respect to an additional reference plane, which reflects the initially developable surface of membrane strips in the production process. Physically problematic combinations of edge geometry and surface stress are solved by a self‐adaptive stress correction scheme. The algorithm is based on a local criterion derived from differential geometry. Several examples illustrate the success of each idea and implementation. Copyright © 2005 John Wiley & Sons, Ltd.     

Non-iterative solution of a tridiagonal-type matrix coupled by diagonal submatrices

A non-iterative method is presented for solving a system of algebraic equations forming a banded matrix with each diagonal submatrix being tridiagonal and all other submatrices being diagonal. A recursion relation is assumed between successive elements of the solution vector. This relation is determined by equating a linearly combined form of it with the original set of equations also being linearly combined. The solution vector is determined by back-substitution of elements in the recursion relation. Since the method takes advantage of the banded structure of the matrix, it is more efficient than the conventional methods. A FORTRAN program that incorporates the present method is included.     

The mode‐III fracture analysis of an inclined crack embedded in thin layer systems by analytical alternating methods

Abstract

This paper analyzes the mode‐III stress intensity factor of an inclined crack, embedded in a thin layer, bonded to a half plane, subjected to arbitrary distributed anti‐plane loads. Special alternating procedures are presented to evaluate the mode‐ III S.I.F. and the numerical results confirm the validity of the proposed alternating procedure. The solution of a bi‐material problem in an infinite plane with an inclined crack and the analytical solution of a thin layer, without crack, bonded to a half plane, subjected to an anti‐plane point force applied on the boundary are referred to as fundamental solutions. By using these fundamental solutions and alternating procedures, the stress intensity factors of a crack in a thin layer bonded to a half plane are evaluated. The numerical results of some reduced problems are computed and excellent agreements with existing solutions are obtained.     

An adaptive algebraic multigrid algorithm for micromagnetism

We present an adaptive algebraic multigrid algorithm. The method is intended for large sparse matrix equations that arise from finite-element discretizations of the stray field in three-dimensional micromagnetism on nonuniform or unstructured grids. It uses a varying threshold value to control the grid ratio, trying to optimize the overall efficiency of the algebraic multigrid solver. We present numerical results and compare them with the preconditioned conjugate gradient method.     

Optimization of large‐scale truss structures using sparse SAND formulations

Sparsity features of simultaneous analysis and design (SAND) formulations are studied and exploited for optimization of large‐scale truss structures. Three formulations are described and implemented with an existing analysis code. SAND formulations have large number of variables; however, gradients of the functions and Hessian of the Lagrangian are quite sparsely populated. Therefore, this structure of the problem is exploited and an optimization algorithm with sparse matrix capability is used to solve large‐scale problems. An existing analysis software is integrated with an optimizer based on a sparse sequential quadratic programming (SQP) algorithm to solve sample problems. The formulations and algorithms work quite well for the sample problems, and their performances are compared and discussed. For all the cases considered, the SAND formulations out perform the conventional formulation except one case. Further research is suggested to fully study and utilize sparse features of the alternative SAND formulations and to develop more efficient sparse solvers for large‐scale and more complex applications. Copyright © 2006 John Wiley & Sons, Ltd.     

Integrating a micromechanical model for multiscale analyses

The Chang‐Hicher micromechanical model based on a static hypothesis, not unlike other models developed separately at around the same epoch, has proved its efficiency in predicting soil behaviour. For solving boundary value problems, the model has now integrated stress‐strain relationships by considering both the micro and macro levels. The first step was to solve the linearized mixed control constraints by the introduction of a predictor–corrector scheme and then to implement the micro–macro relationships through an iterative procedure. Two return mapping schemes, consisting of the closest‐point projection method and the cutting plane algorithm, were subsequently integrated into the interparticle force‐displacement relations. Both algorithms have proved to be efficient in studies devoted to elementary tests and boundary value problems. Closest‐point projection method compared with cutting plane algorithm, however, has the advantage of being more intensive cost efficient and just as accurate in the computational task of integrating the local laws into the micromechanical model. The results obtained demonstrate that the proposed linearized method is capable of performing loadings under stress and strain control. Finally, by applying a finite element analysis with a biaxial test and a square footing, it can be recognized that the Chang‐Hicher micromechanical model performs efficiently in multiscale modelling.     

Interacting circular inhomogeneities in plane elastostatics

Summary This paper provides a general series solution to the problem of interacting circular inhomogeneities in plane elastostatics. The analysis is based upon the use of the complex stress potentials of Muskhelishvili and the Laurent series expansion method. The general forms of the complex potentials are derived explicitly for the circular inhomogeneity problem under arbitrary plane loading. Using the superposition principle, these general expressions were subsequently employed to treat the problem of an infinitely extended matrix containing any number ofarbitrarily located inhomogeneities. The above procedure reduces the problem to a set of linear algebraic equations which are solved with the aid of a perturbation technique. The current method is shown to be capable of yielding approximate closed-form solutions for multiple inhomogeneities, thus providing the explicit dependence of the solution upon the partinent parameters.     

A systematic construction of B‐bar functions for linear and non‐linear mixed‐enhanced finite elements for plane elasticity problems

In a previous paper a modified Hu–Washizu variational formulation has been used to derive an accurate four node plane strain/stress finite element denoted QE2. For the mixed element QE2 two enhanced strain terms are used and the assumed stresses satisfy the equilibrium equations a priori for the linear elastic case. In this paper an alternative approach is discussed. The new formulation leads to the same accuracy for linear elastic problems as the QE2 element; however it turns out to be more efficient in numerical simulations, especially for large deformation problems. Using orthogonal stress and strain functions we derive B̄ functions which avoid numerical inversion of matrices. The B̄ ‐strain matrix is sparse and has the same structure as the strain matrix B obtained from a compatible displacement field. The implementation of the derived mixed element is basically the same as the one for a compatible displacement element. The only difference is that we have to compute a B̄ ‐strain matrix instead of the standard B ‐matrix. Accordingly, existing subroutines for a compatible displacement element can be easily changed to obtain the mixed‐enhanced finite element which yields a higher accuracy than the Q4 and QM6 elements. Copyright © 1999 John Wiley & Sons, Ltd.     

A return‐mapping algorithm for plastic‐damage models: 3‐D and plane stress formulation

Three‐dimensional and plane stress formulations of the return‐mapping algorithm for a class of plastic‐damage models are derived using the spectral decomposition form of the stress. An efficient plane stress computation scheme based on the spectral return‐mapping algorithm is developed. The consistent algorithmic tangent stiffness for the present algorithm is formulated. The validation and performance of the present return‐mapping algorithm is demonstrated by numerical examples. Copyright © 2001 John Wiley & Sons, Ltd.     

Simple and efficient integration of rigid rotations suitable for constraint solvers

Simple and efficient way of integrating rigid rotations is presented. The algorithm is stable, second‐order accurate, and in its explicit version involves evaluation of only two exponential maps per time step. The semi‐explicit version of the proposed scheme improves upon the long‐term stability, while it retains the explicitness in the force evaluation. The algebraic structure of both schemes makes them suitable forthe analysis of constrained multi‐body systems. The explicit algorithm is specifically aimed at the analysis involving small incremental rotations, where its modest computational cost becomes the major advantage. The semi‐explicit scheme naturally broadens the scope of possible applications. Copyright © 2009 John Wiley & Sons, Ltd.     

Cycle bases for the flexibility analysis of structures

Two methods are presented for the automatic selection of a cycle basis leading to a sparse flexibility matrix for the analysis of rigid-jointed skeletal structures. The first method having a local approach forms a maximal set of admissible minimal cycles, while the second having a global approach constructs admissible minimal cycles on the ordered chords of a shortest route tree. A cycle ordering algorithm is also given to reduce the band width of the corresponding flexibility matrix.     

A new mode renumbering algorithm for bandwidth reduction

A new algorithm is presented for automatic renumbering of systems of interconnected nodes so as to minimize the bandwidth of the connectivity matrix. This is necessary to reduce storage requirements for banded matrix solution techniques. The method is based on those due to Cuthill and McKee¹ and Gibbs, Poole and Stockmeyer.² Under test against several other algorithms on a range of 20 examples of various types it always performed at least as well as, and in most cases better than, the best of the other methods. Seven examples are given in the paper, comparing final bandwidths with those produced by seven other algorithms.