摘要:

AbstractThis paper compares the performance of the SS22 and SS23 (References 1 and 2) single step algorithms for the numerical solution of the second‐order structural dynamic equation and a related new algorithm SS32B applied to the equivalent first‐order system, with sine and step forcing functions. Various aspects of stability relevant to these equations are discus

ISSN:0029-5981    

DOI:10.1002/nme.1620211102

出版商:John Wiley&Sons, Ltd    

年代:1985

数据来源: WILEY

摘要:

AbstractA method for obtaining difference equations from differential equations which has previously been applied to ordinary differential equations is here applied to the time‐dependent heat conduction equation as a representative example of parabolic partial differential equations. Explicit difference equations are derived which are stable for all values of αΔt/Δ2and are also accurate. Implicit‐algorithms with improved accuracy are also derived. Problems in cartesian and cylindrical co‐ordinates are

ISSN:0029-5981    

DOI:10.1002/nme.1620211103

出版商:John Wiley&Sons, Ltd    

年代:1985

数据来源: WILEY

摘要:

AbstractIn general, the Lagrange multiplier method (LMM) is used to incorporate subsidiary conditions into variational principles. The Lagrange multipliers represent an additional field of independent variables. The attempt to satisfy subsidiary conditions without employing additional independent unknowns has led to the development of simplified variational principles (SVP). They are characterized by expressing Lagrange multipliers in terms of original field variables by means of the Euler‐Lagrange equations for the multipliers, providing their physical interpretation.In the first part of the theoretical investigation, systems with infinitely many degrees‐of‐freedom are studied. It is shown that the Euler‐Lagrange equations of a LMM based on a modification of the principle of minimum of potential energy (PMIPE) do not hold unconditionally, that is, for arbitrary subsidiary conditions, for the corresponding SVP. The second part of the theoretical investigation is concerned with systems with a finite number of degrees‐of‐freedom. The finite element method (FEM) is employed to discuss and compare the characteristics of the LMM and of the corresponding SVP. Contrary to the former, the latter are found to be problem‐dependent. Several shortcomings of the SVP are listed, including the possibility of obtaining an infinite sequence of singular coefficient matrices in the process of a systematic mesh refinement. Consequently, convergence of finite element solutions to the true solution in the limit of finite element representations is not guaranteed. The theoretical findings are corroborated by the results of a detailed nu

ISSN:0029-5981    

DOI:10.1002/nme.1620211104

出版商:John Wiley&Sons, Ltd    

年代:1985

数据来源: WILEY

摘要:

AbstractA functional in the form of a convolution of the temperature variation in the space–time domain has been derived. It has been used as the basis for a finite element formulation of heat conduction in solids. Numerical illustrations have indicated that the space–time finite element algorithm provides a more rapid convergence to the exact solutions than the usual finite element analysis with discretization in space o

ISSN:0029-5981    

DOI:10.1002/nme.1620211105

出版商:John Wiley&Sons, Ltd    

年代:1985

数据来源: WILEY

摘要:

AbstractAn effective technique is presented for recovering surface tractions and distributed reaction forces from finite element displacement solutions of structural mechanics problems. Nodal values of surface tractions and assumed trial functions are used to represent the distribution of surface stresses. Equations obtained from the familiar Galerkin formulation are used to compute the nodal traction values. The method is demonstrated to be efficient and highly accurate, even in the presence of jump conditions or singularities. Extensions of the method are possible for recovering secondary solution variables in non‐structural finite element application

ISSN:0029-5981    

DOI:10.1002/nme.1620211106

出版商:John Wiley&Sons, Ltd    

年代:1985

数据来源: WILEY

摘要:

AbstractThis paper sets the stage for the implementation of the Fixed‐Point Iteration (FPI) to nonlinear finite element analysis as an alternative to the existing Newton‐Raphson Method (NRM) or its derivatives. The superiority of the former method over the latter one is such that it enables one to obtain nonlinear structural static or dynamic responses without inverting the structural stiffness matrix.In the first part of the paper, a new convergence correction/acceleration factor has been developed for the FPI when applied to a single nonlinear algebraic equation. This new factor causes the iteration function of the equation under consideration to rotate about an axis that passes through one of its fixed points or roots. Using this observation, the slope of the iteration function can be adjusted in the neighbourhood of a specific root to ensure the convergence of the FPI. It is found that the optimum choice of the new factor corresponds to a zero slope, evaluated at the root, of the iteration function. The rate of convergence and the error estimate of this form of the FPI is developed and compared with the NRM. The equilibrium positions of a nonlinear loaded softening spring have been obtained by both methods as an illustrative numerical example to measure the effect of the new factor on the convergence rate.The second part of the paper extends the above concept to find the solution of a linear system of algebraic equations using the FPI. This leads to a better diagonal approximate inverse for the Jacobi iteration, or method of simultaneous displacements. If the elements of the solution vector of a specific system are all equal, the new Jacobi iteration becomes an exact method and the solution is obtained in one iteration. The concept is also extended to the Gauss‐Seidel iteration, or method of successive displacements. Systems involving symmetric as well as nonsymmetric coefficient matrices have been used as numerical examples and are presented. For future implementation to nonlinear finite element analysis, the active column or the skyline (or the non‐zero profile) FPI algorithms are developed for programming conside

ISSN:0029-5981    

DOI:10.1002/nme.1620211107

出版商:John Wiley&Sons, Ltd    

年代:1985

数据来源: WILEY

摘要:

AbstractThe finite element formulation and implementation of the Fixed‐Point Iteration (FPI) to linear/nonlinear structural static or dynamic analysis are developed. The direct and tangent formulations are presented and compared with the Newton–Raphson method (NRM). ‘Modified’ FPI algorithms have also been derived. A graphical interpretation of the method is introduced and suggested to call the FPI ‘the Saw method’. Mixing both the FPI and NRM is shown to be possible and may be useful in some applications. The overall strategies, iterative algorithms, and the appropriate norm convergence criteria necessary to implant the FPI into existing finite element programs are also included in the development.The superiority of the FPI over the NRM as seen from the development and the formulation lies in three major factors. First, the existing assembly process of element matrices is eliminated completely from the nonlinear finite element analysis. Secondly, the Gauss elimination or Crout's method is also eliminated. In the finite element terminology, this means that nonlinear structural static or dynamic responses can he obtained without recourse to the inverse of the structural stiffness matrix. Thirdly, the FPI can also be applied equally tolinearstructural analysis. Hence, the assembly process and the programming and storage associated with it can be removed from the existing finite element programs.While the FPI can solve problems that the NRM can, it will also be able to handle some engineering problems where the latter cannot. Buckling problems and problems where the force–displacement curve changes curvature are examples where the FPI is expected to

ISSN:0029-5981    

DOI:10.1002/nme.1620211108

出版商:John Wiley&Sons, Ltd    

年代:1985

数据来源: WILEY

摘要:

AbstractThe purpose of this paper is to report on a new and efficient method for the evaluation of singular integrals in stress analysis of elastic and elasto‐plastic solids, respectively, by the direct boundary element method (BEM). Triangle polar co‐ordinates are used to reduce the order of singularity of the boundary integrals by one degree and to carry out the integration over mappings of the boundary elements onto plane squares. The method was subsequently extended to the cubature of singular integrals over three‐dimensional internal cells as occur in applications of the BEM to three‐dimensional elasto‐plasticity. For this purpose so‐called tetrahedron polar co‐ordinates were introduced. Singular boundary integrals stretching over either linear, triangular, or quadratic quadilateral, isoparametric boundry elements and singular volume integrals extending over either linear, tetrahedral, or quadratic, hexahedral, isoparametric internal cells are treated. In case of higher order isoparametric boundary elements and internal cells, division into a number of subelements and subcells, respectively, is necessary. The analytical investigation is followed by a numerical study restricted to the use of quadratic, quadrilateral, isoparametric boundary elements. This is justified by the fact that such elements, as opposed to linear elements, yield singular boundary integrals which cannot be integrated analytically. The results of the numerical investigation demonstrate the potential of the deve

ISSN:0029-5981    

DOI:10.1002/nme.1620211109

出版商:John Wiley&Sons, Ltd    

年代:1985

数据来源: WILEY

摘要:

AbstractA general class of time‐dependent co‐ordinate transformations is introduced in a variational formulation for evolution problems. The variational problem is posed with respect to both solution and transformation field variables. An approximate analysis using finite elements is developed from the continuous variational form. Modified forms of the variational functional are considered to ensure the deforming mesh is not top irregular. ODE system integrators are utilized to integrate the resulting semidiscrete systems. In Part I we consider the formulation for problems in one spatial dimension and time, including, in particular, convection‐dominated flows described by the convection‐diffusion, Burgers' and Buckley–Leverett equations. In Part II the extension of the method to two dimensions and supporting numerical experiments are

ISSN:0029-5981    

DOI:10.1002/nme.1620211110

出版商:John Wiley&Sons, Ltd    

年代:1985

数据来源: WILEY

ISSN:0029-5981    

DOI:10.1002/nme.1620211112

出版商:John Wiley&Sons, Ltd    

年代:1985

数据来源: WILEY