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Elasticity Problems (elasticity + problem)

Distribution by Scientific Domains

Engineering	65%
Mathematics and Statistics	34%

Kinds of Elasticity Problems

linear elasticity problem

Selected Abstracts

On computing the forces from the noisy displacement data of an elastic body

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2008
A. Narayana Reddy
Abstract This study is concerned with the accurate computation of the unknown forces applied on the boundary of an elastic body using its measured displacement data with noise. Vision-based minimally intrusive force-sensing using elastically deformable grasping tools is the motivation for undertaking this problem. Since this problem involves incomplete and inconsistent displacement/force of an elastic body, it leads to an ill-posed problem known as Cauchy's problem in elasticity. Vision-based displacement measurement necessitates large displacements of the elastic body for reasonable accuracy. Therefore, we use geometrically non-linear modelling of the elastic body, which was not considered by others who attempted to solve Cauchy's elasticity problem before. We present two methods to solve the problem. The first method uses the pseudo-inverse of an over-constrained system of equations. This method is shown to be not effective when the noise in the measured displacement data is high. We attribute this to the appearance of spurious forces at regions where there should not be any forces. The second method focuses on minimizing the spurious forces by varying the measured displacements within the known accuracy of the measurement technique. Both continuum and frame elements are used in the finite element modelling of the elastic bodies considered in the numerical examples. The performance of the two methods is compared using seven numerical examples, all of which show that the second method estimates the forces with an error that is not more than the noise in the measured displacements. An experiment was also conducted to demonstrate the effectiveness of the second method in accurately estimating the applied forces. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Analytical study and numerical experiments for degenerate scale problems in the boundary element method for two-dimensional elasticity

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2002
J. T. Chen
Abstract For a plane elasticity problem, the boundary integral equation approach has been shown to yield a non-unique solution when geometry size is equal to a degenerate scale. In this paper, the degenerate scale problem in the boundary element method (BEM) is analytically studied using the method of stress function. For the elliptic domain problem, the numerical difficulty of the degenerate scale can be solved by using the hypersingular formulation instead of using the singular formulation in the dual BEM. A simple example is shown to demonstrate the failure using the singular integral equations of dual BEM. It is found that the degenerate scale also depends on the Poisson's ratio. By employing the hypersingular formulation in the dual BEM, no degenerate scale occurs since a zero eigenvalue is not embedded in the influence matrix for any case. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Self-regular boundary integral equation formulations for Laplace's equation in 2-D

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2001
A. B. Jorge
Abstract The purpose of this work is to demonstrate the application of the self-regular formulation strategy using Green's identity (potential-BIE) and its gradient form (flux-BIE) for Laplace's equation. Self-regular formulations lead to highly effective BEM algorithms that utilize standard conforming boundary elements and low-order Gaussian integrations. Both formulations are discussed and implemented for two-dimensional potential problems, and numerical results are presented. Potential results show that the use of quartic interpolations is required for the flux-BIE to show comparable accuracy to the potential-BIE using quadratic interpolations. On the other hand, flux error results in the potential-BIE implementation can be dominated by the numerical integration of the logarithmic kernel of the remaining weakly singular integral. Accuracy of these flux results does not improve beyond a certain level when using standard quadrature together with a special transformation, but when an alternative logarithmic quadrature scheme is used these errors are shown to reduce abruptly, and the flux results converge monotonically to the exact answer. In the flux-BIE implementation, where all integrals are regularized, flux results accuracy improves systematically, even with some oscillations, when refining the mesh or increasing the order of the interpolating function. The flux-BIE approach presents a great numerical sensitivity to the mesh generation scheme and refinement. Accurate results for the potential and the flux were obtained for coarse-graded meshes in which the rate of change of the tangential derivative of the potential was better approximated. This numerical sensitivity and the need for graded meshes were not found in the elasticity problem for which self-regular formulations have also been developed using a similar approach. Logarithmic quadrature to evaluate the weakly singular integral is implemented in the self-regular potential-BIE, showing that the magnitude of the error is dependent only on the standard Gauss integration of the regularized integral, but not on this logarithmic quadrature of the weakly singular integral. The self-regular potential-BIE is compared with the standard (CPV) formulation, showing the equivalence between these formulations. The self-regular BIE formulations and computational algorithms are established as robust alternatives to singular BIE formulations for potential problems. Copyright © 2001 John Wiley & Sons, Ltd. [source]

Flow simulation on moving boundary-fitted grids and application to fluid,structure interaction problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2006
Martin Engel
Abstract We present a method for the parallel numerical simulation of transient three-dimensional fluid,structure interaction problems. Here, we consider the interaction of incompressible flow in the fluid domain and linear elastic deformation in the solid domain. The coupled problem is tackled by an approach based on the classical alternating Schwarz method with non-overlapping subdomains, the subproblems are solved alternatingly and the coupling conditions are realized via the exchange of boundary conditions. The elasticity problem is solved by a standard linear finite element method. A main issue is that the flow solver has to be able to handle time-dependent domains. To this end, we present a technique to solve the incompressible Navier,Stokes equation in three-dimensional domains with moving boundaries. This numerical method is a generalization of a finite volume discretization using curvilinear coordinates to time-dependent coordinate transformations. It corresponds to a discretization of the arbitrary Lagrangian,Eulerian formulation of the Navier,Stokes equations. Here the grid velocity is treated in such a way that the so-called Geometric Conservation Law is implicitly satisfied. Altogether, our approach results in a scheme which is an extension of the well-known MAC-method to a staggered mesh in moving boundary-fitted coordinates which uses grid-dependent velocity components as the primary variables. To validate our method, we present some numerical results which show that second-order convergence in space is obtained on moving grids. Finally, we give the results of a fully coupled fluid,structure interaction problem. It turns out that already a simple explicit coupling with one iteration of the Schwarz method, i.e. one solution of the fluid problem and one solution of the elasticity problem per time step, yields a convergent, simple, yet efficient overall method for fluid,structure interaction problems. Copyright © 2005 John Wiley & Sons, Ltd. [source]

Algebraic multilevel preconditioning of finite element matrices using local Schur complements

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 1 2006
J. K. Kraus
Abstract We consider an algebraic multilevel preconditioning technique for SPD matrices arising from finite element discretization of elliptic PDEs. In particular, we address the case of non-M matrices. The method is based on element agglomeration and assumes access to the individual element matrices. The left upper block of the considered multiplicative two-level preconditioner is approximated using incomplete factorization techniques. The coarse-grid element matrices are simply Schur complements computed from local neighbourhood matrices, i.e. small collections of element matrices. Assembling these local Schur complements results in a global Schur complement approximation that can be analysed by regarding (local) macro elements. These components, when combined in the framework of an algebraic multilevel iteration, yield a robust and efficient linear solver. The presented numerical experiments include also the Lamé differential equation for the displacements in the two-dimensional plane-stress elasticity problem. Copyright © 2005 John Wiley & Sons, Ltd. [source]

On parallel solution of linear elasticity problems.

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 3 2002
Part II: Methods, some computer experiments
Abstract This is the second part of a trilogy on parallel solution of the linear elasticity problem. We consider the plain case of the problem with isotropic material, including discontinuous coefficients, and with homogeneous Dirichlet boundary condition. The discretized problem is solved by the preconditioned conjugate gradient (pcg) method. In the first part of the trilogy block- diagonal preconditioners based on the separate displacement component part of the elasticity equations were analysed. The preconditioning systems were solved by the pcg-method, i.e. inner iterations were performed. As preconditioner, we used modified incomplete factorization MIC(0), where possibly the element matrices were modified in order to give M -matrices, i.e. in order to guarantee the existence of the MIC(0) factorization. In the present paper, the second part, full block incomplete factorization preconditioners are presented and analysed. In order to avoid inner/outer iterations we also study a variant of the block-diagonal method and of the full block method, where the matrices of the inner systems are just replaced by their MIC(0)-factors. A comparison is made between the various methods with respect to rate of convergence and work per unknown. The fastest methods are implemented by message passing utilizing the MPI system. In the third part of the trilogy, we will focus on the use of higher-order finite elements. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Refined mixed finite element method for the elasticity problem in a polygonal domain

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2002
M. Farhloul
Abstract The purpose of this article is to study a mixed formulation of the elasticity problem in plane polygonal domains and its numerical approximation. In this mixed formulation the strain tensor is introduced as a new unknown and its symmetry is relaxed by a Lagrange multiplier, which is nothing else than the rotation. Because of the corner points, the displacement field is not regular in general in the vicinity of the vertices but belongs to some weighted Sobolev space. Using this information, appropriate refinement rules are imposed on the family of triangulations in order to recapture optimal error estimates. Moreover, uniform error estimates in the Lamé coefficient , are obtained for , large. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 323,339, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10009 [source]

A node-based agglomeration AMG solver for linear elasticity in thin bodies

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 3 2009
Prasad S. Sumant
Abstract This paper describes the development of an efficient and accurate algebraic multigrid finite element solver for analysis of linear elasticity problems in two-dimensional thin body elasticity. Such problems are commonly encountered during the analysis of thin film devices in micro-electro-mechanical systems. An algebraic multigrid based on element interpolation is adopted and streamlined for the development of the proposed solver. A new node-based agglomeration scheme is proposed for computationally efficient, aggressive and yet effective generation of coarse grids. It is demonstrated that the use of appropriate finite element discretization along with the proposed algebraic multigrid process preserves the rigid body modes that are essential for good convergence of the multigrid solution. Several case studies are taken up to validate the approach. The proposed node-based agglomeration scheme is shown to lead to development of sparse and efficient intergrid transfer operators making the overall multigrid solution process very efficient. The proposed solver is found to work very well even for Poisson's ratio >0.4. Finally, an application of the proposed solver is demonstrated through a simulation of a micro-electro-mechanical switch. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Computation of a few smallest eigenvalues of elliptic operators using fast elliptic solvers

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 8 2001
Janne Martikainen
Abstract The computation of a few smallest eigenvalues of generalized algebraic eigenvalue problems is studied. The considered problems are obtained by discretizing self-adjoint second-order elliptic partial differential eigenvalue problems in two- or three-dimensional domains. The standard Lanczos algorithm with the complete orthogonalization is used to compute some eigenvalues of the inverted eigenvalue problem. Under suitable assumptions, the number of Lanczos iterations is shown to be independent of the problem size. The arising linear problems are solved using some standard fast elliptic solver. Numerical experiments demonstrate that the inverted problem is much easier to solve with the Lanczos algorithm that the original problem. In these experiments, the underlying Poisson and elasticity problems are solved using a standard multigrid method. Copyright © 2001 John Wiley & Sons, Ltd. [source]

On the optimum support size in meshfree methods: A variational adaptivity approach with maximum-entropy approximants

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 7 2010
Adrian Rosolen
Abstract We present a method for the automatic adaption of the support size of meshfree basis functions in the context of the numerical approximation of boundary value problems stemming from a minimum principle. The method is based on a variational approach, and the central idea is that the variational principle selects both the discretized physical fields and the discretization parameters, here those defining the support size of each basis function. We consider local maximum-entropy approximation schemes, which exhibit smooth basis functions with respect to both space and the discretization parameters (the node location and the locality parameters). We illustrate by the Poisson, linear and non-linear elasticity problems the effectivity of the method, which produces very accurate solutions with very coarse discretizations and finds unexpected patterns of the support size of the shape functions. Copyright © 2009 John Wiley & Sons, Ltd. [source]

An adaptive stabilization strategy for enhanced strain methods in non-linear elasticity

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2010
Alex Ten Eyck
Abstract This paper proposes and analyzes an adaptive stabilization strategy for enhanced strain (ES) methods applied to quasistatic non-linear elasticity problems. The approach is formulated for any type of enhancements or material models, and it is distinguished by the fact that the stabilization term is solution dependent. The stabilization strategy is first constructed for general linearized elasticity problems, and then extended to the non-linear elastic regime via an incremental variational principle. A heuristic choice of the stabilization parameters is proposed, which in the numerical examples proved to provide stable approximations for a large range of deformations, different problems and material models. We also provide explicit lower bounds for the stabilization parameters that guarantee that the method will be stable. These are not advocated, since they are generally larger than the ones based on heuristics, and hence prone to deteriorate the locking-free behavior of ES methods. Numerical examples with two different non-linear elastic models in thin geometries and incompressible situations show that the method remains stable and locking free over a large range of deformations. Finally, the method is strongly based on earlier developments for discontinuous Galerkin methods, and hence throughout the paper we offer a perspective about the similarities between the two. Copyright © 2009 John Wiley & Sons, Ltd. [source]

A hybridizable discontinuous Galerkin method for linear elasticity

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2009
S.-C. Soon
Abstract This paper describes the application of the so-called hybridizable discontinuous Galerkin (HDG) method to linear elasticity problems. The method has three significant features. The first is that the only globally coupled degrees of freedom are those of an approximation of the displacement defined solely on the faces of the elements. The corresponding stiffness matrix is symmetric, positive definite, and possesses a block-wise sparse structure that allows for a very efficient implementation of the method. The second feature is that, when polynomials of degree k are used to approximate the displacement and the stress, both variables converge with the optimal order of k+1 for any k,0. The third feature is that, by using an element-by-element post-processing, a new approximate displacement can be obtained that converges at the order of k+2, whenever k,2. Numerical experiments are provided to compare the performance of the HDG method with that of the continuous Galerkin (CG) method for problems with smooth solutions, and to assess its performance in situations where the CG method is not adequate, that is, when the material is nearly incompressible and when there is a crack. Copyright © 2009 John Wiley & Sons, Ltd. [source]

Axial symmetric elasticity analysis in non-homogeneous bodies under gravitational load by triple-reciprocity boundary element method

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 7 2009
Yoshihiro Ochiai
Abstract In general, internal cells are required to solve elasticity problems by involving a gravitational load in non-homogeneous bodies with variable mass density when using a conventional boundary element method (BEM). Then, the effect of mesh reduction is not achieved and one of the main merits of the BEM, which is the simplicity of data preparation, is lost. In this study, it is shown that the domain cells can be avoided by using the triple-reciprocity BEM formulation, where the density of domain integral is expressed in terms of other fields that are represented by boundary densities and/or source densities at isolated interior points. Utilizing the rotational symmetry, the triple-reciprocity BEM formulation is developed for axially symmetric elasticity problems in non-homogeneous bodies under gravitational force. A new computer program was developed and applied to solve several test problems. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Smooth finite element methods: Convergence, accuracy and properties

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2008
Hung Nguyen-Xuan
Abstract A stabilized conforming nodal integration finite element method based on strain smoothing stabilization is presented. The integration of the stiffness matrix is performed on the boundaries of the finite elements. A rigorous variational framework based on the Hu,Washizu assumed strain variational form is developed. We prove that solutions yielded by the proposed method are in a space bounded by the standard, finite element solution (infinite number of subcells) and a quasi-equilibrium finite element solution (a single subcell). We show elsewhere the equivalence of the one-subcell element with a quasi-equilibrium finite element, leading to a global a posteriori error estimate. We apply the method to compressible and incompressible linear elasticity problems. The method can always achieve higher accuracy and convergence rates than the standard finite element method, especially in the presence of incompressibility, singularities or distorted meshes, for a slightly smaller computational cost. It is shown numerically that the one-cell smoothed four-noded quadrilateral finite element has a convergence rate of 2.0 in the energy norm for problems with smooth solutions, which is remarkable. For problems with rough solutions, this element always converges faster than the standard finite element and is free of volumetric locking without any modification of integration scheme. Copyright © 2007 John Wiley & Sons, Ltd. [source]

Complex variable moving least-squares method: a meshless approximation technique

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2007
K. M. Liew
Abstract Based on the moving least-squares (MLS) approximation, we propose a new approximation method,the complex variable moving least-squares (CVMLS) approximation. With the CVMLS approximation, the trial function of a two-dimensional problem is formed with a one-dimensional basis function. The number of unknown coefficients in the trial function of the CVMLS approximation is less than in the trial function of the MLS approximation, and we can thus select fewer nodes in the meshless method that is formed from the CVMLS approximation than are required in the meshless method of the MLS approximation with no loss of precision. The meshless method that is derived from the CVMLS approximation also has a greater computational efficiency. From the CVMLS approximation, we propose a new meshless method for two-dimensional elasticity problems,the complex variable meshless method (CVMM),and the formulae of the CVMM for two-dimensional elasticity problems are obtained. Compared with the conventional meshless method, the CVMM has a greater precision and computational efficiency. For the purposes of demonstration, some selected numerical examples are solved using the CVMM. Copyright © 2006 John Wiley & Sons, Ltd. [source]

On singularities in the solution of three-dimensional Stokes flow and incompressible elasticity problems with corners

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2004
A. Dimitrov
Abstract In this paper, a numerical procedure is presented for the computation of corner singularities in the solution of three-dimensional Stokes flow and incompressible elasticity problems near corners of various shape. For obtaining the order and mode of singularity, a neighbourhood of the singular point is considered with only local boundary conditions. The weak formulation of this problem is approximated using a mixed u, p Galerkin,Petrov finite element method. Additionally, a separation of variables is used to reduce the dimension of the original problem. As a result, the quadratic eigenvalue problem (P+,Q+,2R)d=0 is obtained, where the saddle-point-type matrices P, Q, R are defined explicitly. For a numerical solution of the algebraic eigenvalue problem an iterative technique based on the Arnoldi method in combination with an Uzawa-like scheme is used. This technique needs only one direct matrix factorization as well as few matrix,vector products for finding all eigenvalues in the interval ,,(,) , (,0.5, 1.0), as well as the corresponding eigenvectors. Some benchmark tests show that this technique is robust and very accurate. Problems from practical importance are also analysed, for instance the surface-breaking crack in an incompressible elastic material and the three-dimensional viscous flow of a Newtonian fluid past a trihedral corner. Copyright © 2004 John Wiley & Sons, Ltd. [source]

A posteriori error approximation in EFG method

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 15 2003
L. Gavete
Abstract Recently, considerable effort has been devoted to the development of the so-called meshless methods. Meshless methods still require considerable improvement before they equal the prominence of finite elements in computer science and engineering. One of the paths in the evolution of meshless methods has been the development of the element free Galerkin (EFG) method. In the EFG method, it is obviously important that the ,a posteriori error' should be approximated. An ,a posteriori error' approximation based on the moving least-squares method is proposed, using the solution, computed from the EFG method. The error approximation procedure proposed in this paper is simple to construct and requires, at most, nearest neighbour information from the EFG solution. The formulation is based on employing different moving least-squares approximations. Different selection strategies of the moving least-squares approximations have been used and compared, to obtain optimum values of the parameters involved in the approximation of the error. The performance of the developed approximation of the error is illustrated by analysing different examples for two-dimensional (2D) potential and elasticity problems, using regular and irregular clouds of points. The implemented procedure of error approximation allows the global energy norm error to be estimated and also provides a good evaluation of local errors. Copyright © 2003 John Wiley & Sons, Ltd. [source]

On the use of Somigliana's formula and Fourier series for elasticity problems with circular boundaries

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2003
S. L. Crouch
Abstract This paper considers the problem of an infinite, isotropic elastic plane containing an arbitrary number of non-overlapping circular holes and isotropic elastic inclusions. The holes and inclusions are of arbitrary size and the elastic properties of all of the inclusions can, if desired, be different. The analysis is based on the two-dimensional version of Somigliana's formula, which gives the displacements at a point inside a region V in terms of integrals of the tractions and displacements over the boundary S of this region. We take V to be the infinite plane, and S to be an arbitrary number of circular holes within this plane. Any (or all) of the holes can contain an elastic inclusion, and we assume for simplicity that all inclusions are perfectly bonded to the material matrix. The displacements and tractions on each circular boundary are represented as truncated Fourier series, and all of the integrals involved in Somigliana's formula are evaluated analytically. An iterative solution algorithm is used to solve the resulting system of linear algebraic equations. Several examples are given to demonstrate the accuracy and efficiency of the numerical method. Copyright © 2003 John Wiley & Sons, Ltd. [source]

A spline strip kernel particle method and its application to two-dimensional elasticity problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2003
K. M. Liew
Abstract In this paper we present a novel spline strip kernel particle method (SSKPM) that has been developed for solving a class of two-dimensional (2D) elasticity problems. This new approach combines the concepts of the mesh-free methods and the spline strip method. For the interpolation of the assumed displacement field, we employed the kernel particle shape functions in the transverse direction, and the B3 -spline function in the longitudinal direction. The formulation is validated on several beam and semi-infinite plate problems. The numerical results of these test problems are then compared with the existing solutions obtained by the exact or numerical methods. From this study we conclude that the SSKPM is a potential alternative to the classical finite strip method (FSM). Copyright © 2003 John Wiley & Sons, Ltd. [source]

On Cauchy estimates and growth orders of entire solutions of iterated Dirac and generalized Cauchy,Riemann equations

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 14 2006
D. Constales
Abstract In this paper, we study the growth behaviour of entire Clifford algebra-valued solutions to iterated Dirac and generalized Cauchy,Riemann equations in higher-dimensional Euclidean space. Solutions to this type of systems of partial differential equations are often called k -monogenic functions or, more generically, polymonogenic functions. In the case dealing with the Dirac operator, the function classes of polyharmonic functions are included as particular subcases. These are important for a number of concrete problems in physics and engineering, such as, for example, in the case of the biharmonic equation for elasticity problems of surfaces and for the description of the stream function in the Stokes flow regime with high viscosity. Furthermore, these equations in turn are closely related to the polywave equation, the poly-heat equation and the poly-Klein,Gordon equation. In the first part we develop sharp Cauchy-type estimates for polymonogenic functions, for equations in the sense of Dirac as well as Cauchy,Riemann. Then we introduce generalizations of growth orders, of the maximum term and of the central index in this framework, which in turn then enable us to perform a quantitative asymptotic growth analysis of this function class. As concrete applications we develop some generalizations of some of Valiron's inequalities in this paper. Copyright © 2006 John Wiley & Sons, Ltd. [source]

Further results on the asymptotic growth of entire solutions of iterated Dirac equations in ,n

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 5 2006
D. Constales
Abstract In this paper, we establish some further results on the asymptotic growth behaviour of entire solutions to iterated Dirac equations in ,n. Solutions to this type of systems of partial differential equations are often called polymonogenic or k -monogenic. In the particular cases where k is even, one deals with polyharmonic functions. These are of central importance for a number of concrete problems arising in engineering and physics, such as for example in the case of the biharmonic equation for the description of the stream function in the Stokes flow regime with low Reynolds numbers and for elasticity problems in plates. The asymptotic study that we are going to perform within the context of these PDE departs from the Taylor series representation of their solutions. Generalizations of the maximum term and the central index serve as basic tools in our analysis. By applying these tools we then establish explicit asymptotic relations between the growth behaviour of polymonogenic functions, the growth behaviour of their iterated radial derivatives and that of functions obtained by applying iterations of the , operator to them. Copyright © 2005 John Wiley & Sons, Ltd. [source]

A geometric-based algebraic multigrid method for higher-order finite element equations in two-dimensional linear elasticity

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 7 2009
Yingxiong Xiao
Abstract In this paper, we will discuss the geometric-based algebraic multigrid (AMG) method for two-dimensional linear elasticity problems discretized using quadratic and cubic elements. First, a two-level method is proposed by analyzing the relationship between the linear finite element space and higher-order finite element space. And then a geometric-based AMG method is obtained with the existing solver used as a solver on the first coarse level. The resulting AMG method is applied to some typical elasticity problems including the plane strain problem with jumps in Young's modulus. The results of various numerical experiments show that the proposed AMG method is much more robust and efficient than a classical AMG solver that is applied directly to the high-order systems alone. Moreover, we present the corresponding theoretical analysis for the convergence of the proposed AMG algorithms. These theoretical results are also confirmed by some numerical tests. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Eigenvalue estimates for preconditioned saddle point matrices

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 4 2006
Owe Axelsson
Abstract New accurate eigenvalue bounds for symmetric matrices of saddle point form are derived and applied for both unpreconditioned and preconditioned versions of the matrices. The estimates enable a better understanding of how preconditioners should be chosen. The preconditioners provide efficient iterative solution of the corresponding linear systems with, for some important applications, an optimal order of computational complexity. The methods are applied for Stokes problem and for linear elasticity problems. Copyright © 2005 John Wiley & Sons, Ltd. [source]

Using the modified 2nd order incomplete Cholesky decomposition as the conjugate gradient preconditioning

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 6-7 2002
I. E. Kaporin
Abstract In this paper, the ,second-order' incomplete triangular factorization (Kaporin, 1998) is considered as a preconditioner for the CG method. Some refinements of the original algorithm are proposed and investigated, which give rise to a more efficient modified incomplete Cholesky 2nd-order (MIC2) type preconditionings. Numerical results are given for a set of real-life large-scale SPD linear systems arising in the finite element modelling of linear elasticity problems which clearly indicate the superiority of the MIC2 preconditionings. Copyright © 2002 John Wiley & Sons, Ltd. [source]

On parallel solution of linear elasticity problems.

A parallel block overlap preconditioning with inexact submatrix inversion for linear elasticity problems

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 2 2002
Igor E. Kaporin
We present a parallel preconditioned iterative solver for large sparse symmetric positive definite linear systems. The preconditioner is constructed as a proper combination of advanced preconditioning strategies. It can be formally seen as being of domain decomposition type with algebraically constructed overlap. Similar to the classical domain decomposition technique, inexact subdomain solvers are used, based on incomplete Cholesky factorization. The proper preconditioner is shown to be near optimal in minimizing the so-called K -condition number of the preconditioned matrix. The efficiency of both serial and parallel versions of the solution method is illustrated on a set of benchmark problems in linear elasticity. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Generalization of robustness test procedure for error estimators.

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2005
Part I: formulation for patches near kinked boundaries
Abstract In this part of paper we shall extend the formulation proposed by Babu,ka and co-workers for robustness patch test, for quality assessment of error estimators, to more general cases of patch locations especially in three-dimensional problems. This is performed first by finding an asymptotic finite element solution at interior parts of a problem with assumed smooth exact solution and then adding a correction part to obtain the solution near a kinked boundary irrespective of other boundary conditions at far ends of the domain. It has been shown that the solution corresponding to the correction part may be obtained in a spectral form by assuming a suitable proportionality relation between the nodal values of a mesh with repeatable pattern of macro-patches. Having found the asymptotic finite element solution, the performance of error estimators may be examined. Although in this paper we focus on the asymptotic behaviour of error estimators, the method described in this part may be used to obtain finite element solution for two/three-dimensional unbounded heat/elasticity problems with homogeneous differential equations. Some numerical results are presented to show the validity and performance of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd. [source]