Showing papers in "International Journal for Numerical Methods in Engineering in 1995"
TL;DR: Numerical and theoretical results show the proposed reproducing kernel interpolation functions satisfy the consistency conditions and the critical time step prediction; furthermore, the RKPM provides better stability than Smooth Particle Hydrodynamics (SPH) methods.
Abstract: This paper explores a Reproducing Kernel Particle Method (RKPM) which incorporates several attractive features. The emphasis is away from classical mesh generated elements in favour of a mesh free system which only requires a set of nodes or particles in space. Using a Gaussian function or a cubic spline function, flexible window functions are implemented to provide refinement in the solution process. It also creates the ability to analyse a specific frequency range in dynamic problems reducing the computer time required. This advantage is achieved through an increase in the critical time step when the frequency range is low and a large window is used. The stability of the window function as well as the critical time step formula are investigated to provide insight into RKPMs. The predictions of the theories are confirmed through numerical experiments by performing reconstructions of given functions and solving elastic and elastic–plastic one-dimensional (1-D) bar problems for both small and large deformation as well as three 2-D large deformation non-linear elastic problems. Numerical and theoretical results show the proposed reproducing kernel interpolation functions satisfy the consistency conditions and the critical time step prediction; furthermore, the RKPM provides better stability than Smooth Particle Hydrodynamics (SPH) methods. In contrast with what has been reported in SPH literature, we do not find any tensile instability with RKPMs.
794 citations
TL;DR: In this paper, a Lagrangian finite element model of orthogonal high-speed machining is developed, which accounts for dynamic effects, heat conduction, mesh-on-mesh contact with friction, and full thermo-mechanical coupling.
Abstract: A Lagrangian finite element model of orthogonal high-speed machining is developed. Continuous remeshing and adaptive meshing are the principal tools which we employ for sidestepping the difficulties associated with deformation-induced element distortion, and for resolving fine-scale features in the solution. The model accounts for dynamic effects, heat conduction, mesh-on-mesh contact with friction, and full thermo-mechanical coupling. In addition, a fracture model has been implemented which allows for arbitrary crack initiation and propagation in the regime of shear localized chips. The model correctly exhibits the observed transition from continuous to segmented chips with increasing tool speed.
529 citations
TL;DR: In this article, a non-linear shell theory is introduced, which provides a complete three-dimensional state of stress, and is applied to quadrilateral shell elements, which provide only displacement degrees of freedom located at nodes on the outer surfaces and one degree of freedom at the middle surface.
Abstract: The paper introduces a non-linear shell theory, which provides a complete three-dimensional state of stress. Since the theory is derived from simple three-dimensional continuum mechanics, it is very easy to understand. As an example, the theory is applied to quadrilateral shell elements, which provide only displacement degrees of freedom located at nodes on the outer surfaces and one degree of freedom at the middle surface. It is proposed to eliminate this degree of freedom on element level, so that the elements have the same layout as the equivalent brick elements, but have a better behaviour in bending, have stress resultants and are cheaper with respect to computational effort. The advantages with respect to implementation in a finite element program, as well as in special applications, are obvious. However, well-known conditioning problems in thin shell applications must be expected. Therefore emphasis is put on this issue in the example problems. It is shown that the elements can give acceptable answers in engineering applications and offer a potential for material non-linear applications, which will be considered in a forthcoming paper.
292 citations
TL;DR: In this paper, the long-term dynamic response of non-linear geometrically exact rods undergoing finite extension, shear and bending, accompanied by large overall motions, is addressed in detail.
Abstract: The long-term dynamic response of non-linear geometrically exact rods under-going finite extension, shear and bending, accompanied by large overall motions, is addressed in detail. The central objective is the design of unconditionally stable time-stepping algorithms which exactly preserve fundamental constants of the motion such as the total linear momentum, the total angular momentum and, for the Hamiltonian case, the total energy. This objective is accomplished in two steps. First, a class of algorithms is introduced which conserves linear and angular momentum. This result holds independently of the definition of the algorithmic stress resultants. Second, an algorithmic counterpart of the elastic constitutive equations is developed such that the law of conservation of total energy is exactly preserved. Conventional schemes exhibiting no numerical dissipation, symplectic algorithms in particular, are shown to lead to unstable solutions when the high frequencies are not resolved. Compared to conventional schemes there is little, if any, additional computational cost involved in the proposed class of energy–momentum methods. The excellent performance of the new algorithm in comparison to other standard schemes is demonstrated in several numerical simulations.
253 citations
TL;DR: In this article, the phase difference between the exact and the numerical solutions is investigated and discussed in the context of other work directed to this topic, and it is shown that previous error estimates in H1-norm are of nondispersive character but hold for medium or high wavenumber on extremely refined mesh only.
Abstract: When applying numerical methods for the computation of stationary waves from the Helmholtz equation, one obtains ‘numerical waves’ that are dispersive also in non-dispersive media. The numerical wave displays a phase velocity that depends on the parameter k of the Helmholtz equation. In dispersion analysis, the phase difference between the exact and the numerical solutions is investigated. In this paper, the authors' recent result on the phase difference for one-dimensional problems is numerically evaluated and discussed in the context of other work directed to this topic. It is then shown that previous error estimates in H1-norm are of nondispersive character but hold for medium or high wavenumber on extremely refined mesh only. On the other hand, recently proven error estimates for constant resolution contain a pollution term. With certain assumptions on the exact solution, this term is of the order of the phase difference. Thus a link is established between the results of dispersion analysis and the results of numerical analysis. Throughout the paper, the presentation and discussion of theoretical results is accompanied by numerical evaluation of several model problems. Special attention is given to the performance of the Galerkin method with a higher order of polynomial approximation p(h-p-version).
251 citations
TL;DR: In this paper, a higher-order compact scheme that is O(h4) on the nine-point 2D stencil is formulated for the steady stream-function vorticity form of the Navier-Stokes equations.
Abstract: A higher-order compact scheme that is O(h4) on the nine-point 2-D stencil is formulated for the steady stream-function vorticity form of the Navier-Stokes equations. The resulting stencil expressions are presented and hence this new scheme can be easily incorporated into existing industrial software. We also show that special treatment of the wall boundary conditions is required. The method is tested on representative model problems and compares very favourably with other schemes in the literature.
239 citations
TL;DR: The selection of the GLS mesh parameter for two dimensions is considered, and leads to elements that exhibit improved phase accuracy, and an optimal GLS parameter is found which reduces phase error for all possible wave vector orientations over elements.
Abstract: In this paper a Galerkin least-squares (GLS) finite element method, in which residuals in least-squares form are added to the standard Galerkin variational equation, is developed to solve the Helmholtz equation in two dimensions. An important feature of GLS methods is the introduction of a local mesh parameter that may be designed to provide accurate solutions with relatively coarse meshes. Previous work has accomplished this for the one-dimensional Helmholtz equation using dispersion analysis. In this paper, the selection of the GLS mesh parameter for two dimensions is considered, and leads to elements that exhibit improved phase accuracy. For any given direction of wave propagation, an optimal GLS mesh parameter is determined using two-dimensional Fourier analysis. In general problems, the direction of wave propagation will not be known a priori. In this case, an optimal GLS parameter is found which reduces phase error for all possible wave vector orientations over elements. The optimal GLS parameters are derived for both consistent and lumped mass approximations. Several numerical examples are given and the results compared with those obtained from the Galerkin method. The extension of GLS to higher-order quadratic interpolations is also presented.
239 citations
TL;DR: Theoretical and computational aspects of vector-like parametrization of three-dimensional finite rotations, which uses only three rotation parameters, are examined in detail in this paper.
Abstract: Theoretical and computational aspects of vector-like parametrization of three-dimensional finite rotations, which uses only three rotation parameters, are examined in detail in this work. The relationship of the proposed parametrization with the intrinsic representation of finite rotations (via an orthogonal matrix) is clearly identified. Careful considerations of the consistent linearization procedure pertinent to the proposed parametrization of finite rotations are presented for the chosen model problem of Reissner's non-linear beam theory. Pertaining details of numerical implementation are discussed for the simplest choice of the finite element interpolations for a 2-node three-dimensional beam element. A number of numerical simulations in three-dimensional finite rotation analysis are presented in order to illustrate the proposed approach.
237 citations
TL;DR: In this article, a framework for damage mechanics of brittle solids is presented and exploited in the design and numerical implementation of an anisotropic model for the tensile failure of concrete.
Abstract: A framework for damage mechanics of brittle solids is presented and exploited in the design and numerical implementation of an anisotropic model for the tensile failure of concrete. The key feature exploited in the analysis is the hypothesis of maximum dissipation, which specifies a unique damage rule for the elastic moduli of the solid once a failure surface is specified. A complete algorithmic treatment of the resulting model is given which renders a useful tool for large-scale inelastic finite element calculations. A rather simple three-surface failure model for concrete, containing essentially no adjustable parameters, is shown to produce results in remarkably good agreement with sample experimental data.
214 citations
180 citations
IMEC1
TL;DR: In this paper, the authors present a computational model for the topology optimization problem of a 2D linear-elastic solid subjected to thermal loads, with a compliance objective function and an isoperimetric constraint on volume.
Abstract: This paper presents the development of a computational model for the topology optimization problem, using a material distribution approach, of a 2-D linear-elastic solid subjected to thermal loads, with a compliance objective function and an isoperimetric constraint on volume. Defining formally the augmented Lagrangian associated with the optimization problem, the optimality conditions are derived analytically. The results of analysis are implemented in a computer code to produce numerical solutions for the optimal topology, considering the temperature distribution independent of design. The design optimization problem is solved via a sequence of linearized subproblems. The computational model developed is tested in example problems. The influence of both the temperature and the finite element model on the optimal solution obtained is analysed.
TL;DR: The foundations of a new hierarchical modal basis suitable for high-order (hp) finite element discretizations on unstructured meshes is described, based on a generalized tensor product of mixed-weight Jacobi polynomials.
Abstract: In this paper we describe the foundations of a new hierarchical modal basis suitable for high-order (hp) finite element discretizations on unstructured meshes. It is based on a generalized tensor product of mixed-weight Jacobi polynomials. The generalized tensor product property leads to a low operation count with the use of sum factorization techniques. Variable p-order expansions in each element are readily implemented which is a crucial property for efficient adaptive discretizations. Numerical examples demonstrate the exponential convergence for smooth solutions and the ability of this formulation to handle easily very complex two- and three-dmensional computational domains employing standard meshes.
TL;DR: In this paper, the authors studied the pollution-error in the h-version of the finite element method and its effect on the quality of the local error indicators in the interior of the mesh.
Abstract: : We studied the pollution-error in the h-version of the finite element method and its effect on the quality of the local error indicators (resp. the quality of the derivatives recovered by local postprocessing) in the interior of the mesh. Here we show that it is possible to construct a-posteriori estimates of the pollution-error in a patch of elements by employing the local error indicators over the entire mesh. We also give an adaptive algorithm for the local control of the pollution-error in a patch of elements of interest.
TL;DR: In this article, a finite volume (FV) procedure is described for solving the elastic solid mechanics equations in three dimensions on an unstructured mesh, for bodies undergoing thermal or mechanical loads.
Abstract: A Finite Volume (FV) procedure is described for solving the elastic solid mechanics equations in three dimensions on an unstructured mesh, for bodies undergoing thermal or mechanical loads. The FV procedure is developed in parallel with the conventional FE Galerkin procedure so that the differences in each approach may be clearly distinguished. The matrix form of the FV procedure is described, and is implemented in parallel with the FE procedure, both for two-dimensional quadrilateral and three-dimensional brick meshes. The FV and FE procedures are then compared against a range of benchmark problems that test the basic capability of the FV technique. It is shown to be approximately as accurate as the FE procedure on similar meshes, though its system matrix set-up time is twice as long for a node by node set-up procedure.
TL;DR: It is proved that in the limit for large time steps, the new method converges toward the FETI algorithm for time-independent problems, and it is shown that this new domain decomposition method outperforms the popular direct skyline solver.
Abstract: We present a new efficient and scalable domain decomposition method for solving implicitly linear and non-linear time-dependent problems in computational mechanics. The method is derived by adding a coarse problem to the recently proposed transient FETI substructuring algorithm in order to propagate the error globally and accelerate convergence. It is proved that in the limit for large time steps, the new method converges toward the FETI algorithm for time-independent problems. Computational results confirm that the optimal convergence properties of the time-independent FETI method are preserved in the time-dependent case. We employ an iterative scheme for solving efficiently the coarse problem on massively parallel processors, and demonstrate the effective scalability of the new transient FETI method with the large-scale finite element dynamic analysis on the Paragon XP/S and IBM SP2 systems of several diffraction grating finite element structural models. We also show that this new domain decomposition method outperforms the popular direct skyline solver. The coarse problem presented herein is applicable and beneficial to a large class of Lagrange multiplier based substructuring algorithms for time-dependent problems, including the fictitious domain decomposition method.
TL;DR: Simulation results indicate that, in several instances, the optimum solutions obtained using simulated annealing outperform the optimum Solutions obtained using some gradient-based and discrete optimization techniques.
Abstract: A multivariable optimization technique based on the Monte-Carlo method used in statistical mechanics studies of condensed systems is adapted for solving single and multiobjective structural optimization problems. This procedure, known as simulated annealing, draws an analogy between energy minimization in physical systems and objective function minimization in structural systems. The search for a minimum is simulated by a relaxation of the statistical mechanical system where a probabilistic acceptance criterion is used to accept or reject candidate designs. To model the multiple objective functions in the problem formulation, a cooperative game theoretic approach is used. Numerical results obtained using three different annealing strategies for the single and multiobjective design of structures with discrete-continuous variables are presented. The influence of cooling schedule parameters on the optimum solutions obtained is discussed. Simulation results indicate that, in several instances, the optimum solutions obtained using simulated annealing outperform the optimum solutions obtained using some gradient-based and discrete optimization techniques. The results also indicate that simulated annealing has substantial potential for additional applications in optimization, especially for problems with mixed discrete-continuous variables.
TL;DR: In this article, an extension of recent work on the simultaneous optimization of material and structure to address the design of structures under multiple loading conditions is presented, where material properties are represented as elements of the unrestricted set of positive-semi-definite constitutive tensors of a linearly elastic continuum.
Abstract: An extension of recent work 1 on the simultaneous optimization of material and structure to address the design of structures under multiple loading conditions is presented. Material properties are represented in the most general form possible, namely, as elements of the unrestricted set of positive-semi-definite constitutive tensors of a linearly elastic continuum. Existence of solutions can be shown when the objective is a weighted average of compliances and a resource constraint measured as the 2-norm or the trace of the constitutive tensors is included. The optimized material properties can be derived analytically. The optimization of the layout of the material leads to a sizing problem of structural optimization involving a non-linear, non-smooth elasticity analysis. The computational solution of this problem is discussed and illustrated with examples
TL;DR: A new method for analysing plate and shell structures with two or more independently modelled finite element subdomains is presented and it is shown that the hybrid variational formulation provides the most accurate solutions.
Abstract: A new method for analysing plate and shell structures with two or more independently modelled finite element subdomains is presented, assessed, and demonstrated. This method provides a means of coupling local and global finite element models whose nodes do not coincide along their common interface. In general, the method provides a means of coupling structural components (e.g., wing and fuselage) which may have been modelled by different analysts. In both cases, the need for transition modelling, which is often tedious and complicated, is eliminated. The coupling is accomplished through an interface for which three formulations are considered and presented. These formulations are: collocation, discrete least-squares, and hybrid variational. Several benchmark problems are analysed and it is shown that the hybrid variational formulation provides the most accurate solutions.
TL;DR: In this article, an embedded representation of fracture for finite element analysis of concrete structures is presented, where the three-field Hu-Washizu variational statement is extended to bodies with internal discontinuities.
Abstract: As an alternative to the smeared and discrete crack representations, an embedded representation of fracture for finite element analysis of concrete structures is presented. The three-field Hu–Washizu variational statement is extended to bodies with internal discontinuities. The extended variational statement is then utilized for formulating elements with a discontinuous displacement field. The new elements are capable of modelling different deformation modes of an internal discontinuity at the element level. The satisfactory performance of the embedded crack representation is verified by several case studies on concrete fracture.
TL;DR: The large-scale BE capacity provided by this algorithm will not only prove to be useful in large macroscopic models but it will also make it possible to model microscopic damage processes that form the fundamental mechanisms in plastic flow and brittle fracture.
Abstract: In this paper we introduce a method to reduce the solution cost for Boundary Element (BE) models from O(N3)operations to O(N2logN) operations (where N is the number of elements in the model). Previous attempts to achieve such an improvement in efficiency have been restricted in their applicability to problems with regular geometries defined on a uniform mesh. We have developed the Spectral Multipole Method (SMM) which can be used not only for problems with arbitrary geometries but also with a variety of element types. The memory necessary to store the required influence coefficients for the spectral multipole method is O(N) whereas the memory required for the traditional Boundary Element method is O(N2). We demonstrate the savings in computational speed and fast memory requirements in some numerical examples. We have established that the break-even point for the method can be as low as 500 elements, which implies that the method is not only suitable for extremely large-scale problems, but that it also provides a useful bridge between the small-scale and large-scale problems. We also demonstrate the performance of the multipole algorithm on the solution of large-scale granular assembly models. The large-scale BE capacity provided by this algorithm will not only prove to be useful in large macroscopic models but it will also make it possible to model microscopic damage processes that form the fundamental mechanisms in plastic flow and brittle fracture.
TL;DR: In this article, the Lagrange multiplier technique is used to enforce the kinematic constraints among the various bodies of the system, and the forces of constraint are discretized so that the work they perform vanishes exactly.
Abstract: This paper is concerned with the modelling of nonlinear elastic multi-body systems discretized using the finite element method. The formulation uses Cartesian co-ordinates to represent the position of each elastic body with respect to a single inertial frame. The kinematic constraints among the various bodies of the system are enforced via the Lagrange multiplier technique. The resulting equations of motion are stiff, non-linear, differential-algebraic equations. The integration of these equations presents a real challenge as most available techniques are either numerically unstable, or present undesirable high frequency oscillations of a purely numerical origin. An approach is proposed in which the equations of motion are discretized so that they imply conservation of the total energy for the elastic components of the system, whereas the forces of constraint are discretized so that the work they perform vanishes exactly. The combination of these two features of the discretization guarantees the stability of the numerical integration process for non-linear elastic multi-body systems. Examples of the procedure are presented.
TL;DR: In this paper, an assumed strain finite element model with six degrees of freedom per node designed for geometrically non-linear shell analysis is presented, where the kinematics of deformation is described by using vector components in contrast to the conventional formulation which requires the use of trigonometric functions of rotational angles.
Abstract: The present paper describes an assumed strain finite element model with six degrees of freedom per node designed for geometrically non-linear shell analysis. An important feature of the present paper is the discussion on the spurious kinematic modes and the assumed strain field in the geometrically non-linear setting. The kinematics of deformation is described by using vector components in contrast to the conventional formulation which requires the use of trigonometric functions of rotational angles. Accordingly, converged solutions can be obtained for load or displacement increments that are much larger than possible with the conventional formulation with rotational angles. In addition, a detailed study of the spurious kinematic modes and the choice of assumed strain field reveals that the same assumed strain field can be used for both geometrically linear and non-linear cases to alleviate element locking while maintaining kinematic stability. It is strongly recommended that the element models, described in the present paper, be used instead of the conventional shell element models that employ rotational angles.
TL;DR: The parameters used in the algorithm are characterized, and strategies that may be used to improve robustness are suggested, and a method whereby structured tetrahedral meshes with exceptionally stretched elements adjacent to boundary surfaces may be produced.
Abstract: This paper deals with some aspects of unstructured mesh generation in three dimensions by the advancing front technique. In particular, the parameters used in the algorithm are characterized, and strategies that may be used to improve robustness are suggested. We also describe a method whereby structured tetrahedral meshes with exceptionally stretched elements adjacent to boundary surfaces may be produced. The suggested method can be combined with the advancing front concept in a natural way.
TL;DR: The importance of the subdomain aspect ratio as a mesh partitioning factor is emphasized, and its impact on the convergence rate of an optimal domain decomposition based iterative method is highlighted.
Abstract: Optimal domain decomposition methods have emerged as powerful iterative algorithms for parallel implicit computations. Their key preprocessing step is mesh partitioning, where research has focused so far on the automatic generation of load-balanced subdomains with minimum interface nodes. In this paper, we emphasize the importance of the subdomain aspect ratio as a mesh partitioning factor, and highlight its impact on the convergence rate of an optimal domain decomposition based iterative method. We also present a fast optimization algorithm for improving the aspect ratio of existing mesh partitions, and illustrate it with several examples from fluid dynamics and structural mechanics applications. For a stiffened shell problem decomposed by the optimal Recursive Spectral Bisection scheme and solved by the FETI method, this optimization algorithm is shown to improve the solution time by a factor equal to 1·54 and to restore numerical scalability.
TL;DR: In this paper, a class of plasticity models which utilize Rankine's (principal stress) yield locus is formulated to simulate cracking in concrete and rock under monotonic loading conditions.
Abstract: SUMMARY A class of plasticity models which utilize Rankine’s (principal stress) yield locus is formulated to simulate cracking in concrete and rock under monotonic loading conditions. The formulation encompasses isotropic and kinematic hardenindsoftening rules, and incremental (flow theory) as well as total (deformation theory) formats are considered. An Euler backward algorithm is used to integrate the stresses and internal variables over a finite loading step and an explicit expression is derived for a consistently linearized tangent stiffness matrix associated with the Euler backward scheme. Particular attention is paid to the corner regime, that is when the two major principal stresses become equal. A detailed comparison has been made of the proposed plasticity-based crack formulations and the traditional fixed and rotating smeared-crack models for a homogeneously stressed sample under a non-proportional loading path. A comparison between the flow-theory-based plasticity crack models and experimental data has been made for a Single Edge Notched plain concrete specimen under mixed-mode loading conditions.
TL;DR: A finite element formulation based on the work of Yarnada and Okumura 14 is presented to determine the order of singularity and angular variation of the stress and displacement fields surrounding a singular point on a free edge of anisotropic materials as discussed by the authors.
Abstract: A finite element formulation based on the work of Yarnada and Okumura 14 is presented to determine the order of singularity and angular variation of the stress and displacement fields surrounding a singular point on a free edge of anisotropic materials. Emphasis is placed on the computational aspects of this method when applied to configurations including fully bonded multi-material junctions intersecting a free edge as well as materials containing cracks intersecting a free edge. The study shows that the singularity of the three-dimensional stress field may be accurately determined with a relatively small number of elements only when a proper level of numerical integration is used. The method is applied to isotropic and orthotropic materials with a crack intersecting a free edge and an anisotropic three-material junction intersecting a free edge. The efficiency and accuracy of the method indicates it could be used to develop a numerical solution for the singular field that could in turn be used to create free-edge enriched finite elements.
TL;DR: In this article, an improved plane strain/stress element is derived using a Hu-Washizu variational formulation with bilinear displacement interpolation, seven strain and stress terms, and two enhanced strain modes.
Abstract: SUMMARY An improved plane strain/stress element is derived using a Hu-Washizu variational formulation with bilinear displacement interpolation, seven strain and stress terms, and two enhanced strain modes. The number of unknowns of the four-node element is increased from eight to ten degrees of freedom. For linear and non-linear applications, the two unknowns associated with the enhanced strain terms can be eliminated by static condensation so that eight displacement degrees of freedom remain for the proposed element, which is denoted by QE2. The excellent performance of the proposed element is demonstrated using several linear and non-linear examples.
TL;DR: In this article, an augmentation technique is proposed which takes into account micro-mechanical effects, and permits the symmetrization of the tangent stiffness during frictional slip phase.
Abstract: The detailed discretization of contact zones with contact stiffness based on real physical characteristics of contact surfaces can produce stiffness terms which induce ill-conditioning of the global stiffness matrix. Moreover the consistent treatment of frictional behaviour generates non-symmetric tangent stiffness matrices due to the non-associativity of the slip phase. Other non-symmetries are due to the coupling terms and to the dependencies on various parameters that can be involved. To overcome these difficulties almost consistent techniques based on two-step algorithms have been proposed in the past. Here an augmentation technique is proposed which takes into account micro-mechanical effects, and permits the symmetrization of the tangent stiffness during frictional slip phase.
TL;DR: In this paper, the quasi-static and dynamic responses of a linear viscoelastic beam are solved numerically by using the hybrid Laplace transform/finite element method.
Abstract: The quasi-static and dynamic responses of a linear viscoelastic beam are solved numerically by using the hybrid Laplace transform/finite element method In the analysis, the Timoshenko beam theory, which includes the transverse shear and rotatory inertia effect and conventional beam theory, are used to solve this problem The temperature field is assumed to be constant and homogeneous and that the relaxation modulus has the form of the Prony series In the hybrid method, the Laplace transform with respect to time is applied to the coupled equations and the finite element model is developed by applying Hamilton's variational principle without any integral transformation The numerical results of quasi-static and dynamic responses for the models of Maxwell fluid and three parameter solid types are presented and discussed
TL;DR: In this article, a numerical method is described for the computation of eigenpairs which characterize the exact solution of linear second-order elliptic partial differential equations in two dimensions in the vicinity of singular points.
Abstract: A numerical method is described for the computation of eigenpairs which characterize the exact solution of linear second-order elliptic partial differential equations in two dimensions in the vicinity of singular points. The singularities may be caused by re-entrant corners and abrupt changes in boundary conditions or material properties.
Such singularities are of great interest from the point of view of failure initiation: The eigenpairs characterize the straining modes and their amplitudes quantify the amount of energy residing in particular straining modes. For this reason, failure theories directly or indirectly involve the eigenpairs and their amplitudes.
This paper addresses the problem of determining the eigenpairs numerically on the basis of the Steklov formulation. Numerical results are presented for several cases. Importantly, the method is applicable to three-dimensional cases.