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Linear Elasticity (linear + elasticity)
Terms modified by Linear Elasticity Selected AbstractsImproving realism of a surgery simulator: linear anisotropic elasticity, complex interactions and force extrapolationCOMPUTER ANIMATION AND VIRTUAL WORLDS (PREV: JNL OF VISUALISATION & COMPUTER ANIMATION), Issue 3 2002Guillaume Picinbono Abstract In this article, we describe the latest developments of the minimally invasive hepatic surgery simulator prototype developed at INRIA. The goal of this simulator is to provide a realistic training test bed to perform laparoscopic procedures. Therefore, its main functionality is to simulate the action of virtual laparoscopic surgical instruments for deforming and cutting tridimensional anatomical models. Throughout this paper, we present the general features of this simulator including the implementation of several biomechanical models and the integration of two force-feedback devices in the simulation platform. More precisely, we describe three new important developments that improve the overall realism of our simulator. First, we have developed biomechanical models, based on linear elasticity and finite element theory, that include the notion of anisotropic deformation. Indeed, we have generalized the linear elastic behaviour of anatomical models to ,transversally isotropic' materials, i.e. materials having a different behaviour in a given direction. We have also added to the volumetric model an external elastic membrane representing the ,liver capsule', a rather stiff skin surrounding the liver, which creates a kind of ,surface anisotropy'. Second, we have developed new contact models between surgical instruments and soft tissue models. For instance, after detecting a contact with an instrument, we define specific boundary constraints on deformable models to represent various forms of interactions with a surgical tool, such as sliding, gripping, cutting or burning. In addition, we compute the reaction forces that should be felt by the user manipulating the force-feedback devices. The last improvement is related to the problem of haptic rendering. Currently, we are able to achieve a simulation frequency of 25,Hz (visual real time) with anatomical models of complex geometry and behaviour. But to achieve a good haptic feedback requires a frequency update of applied forces typically above 300,Hz (haptic real time). Thus, we propose a force extrapolation algorithm in order to reach haptic real time. Copyright © 2002 John Wiley & Sons, Ltd. [source] An internally consistent integration method for critical state models,INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 3 2005Randall J. Hickman Abstract A new procedure to integrate critical state models including Cam,Clay and modified Cam,Clay is proposed here. The proposed procedure makes use of the linearity of the virgin isotropic compression curve and the parallel anisotropic consolidation lines in e,ln p space which are basic features of the formulation of critical state models. Using this algorithm, a unique final stress state may be found as a function of a single unknown for elastoplastic loading. The key equations are given in this article for the Cam,Clay and modified Cam,Clay models. The use of the Newton,Raphson iterative method to minimize residuals and obtain a converged solution is described here. This new algorithm may be applied using the assumptions of linear elasticity or non-linear elasticity within a given loading step. The new algorithm proposed here is internally consistent and has computational advantages over the current numerical integration procedures. Numerical examples are presented to show the performance of the algorithm as compared to other integration algorithms. Published in 2005 by John Wiley & Sons, Ltd. [source] Determination of rock mass strength properties by homogenizationINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 13 2001A. Pouya Abstract A method for determining fractured rock mass properties is presented here on the basis of homogenization approach. The rock mass is considered to be a heterogeneous medium composed of intact rock and of fractures. Its constitutive model is studied numerically using finite element method and assimilating the fractures to joint elements (Coste, Comportement Thermo-Hydro-Mécanique des massifs rocheux fracturés. Thèse de Doctorat, Ecole Nationale des Ponts et Chaussées, Paris, 1997). The method has been applied to a granite formation in France. Geological data on different families of fractures have been used for the statistical representation of the fractures. A mesh-generating tool for the medium with high density of fractures has been developed. The mechanical behaviour of the rock mass (elasticity, ultimate strength and hardening law) has been determined assuming linear elasticity and Mohr,Coulomb strength criterion both for the intact rock and the fractures. Evolution of the mechanical strength in different directions has been determined as a function of the mean stress, thanks to various numerical simulations. The mechanical strength appears to be anisotropic due to the preferential orientation of the fractures. The numerical results allowed us to determine an oriented strength criterion for the homogenized rock mass. A 2D constitutive law for the homogenized medium has been deduced from numerical data. A 3D extension of this model is also presented. Copyright © 2001 John Wiley & Sons, Ltd. [source] A node-based agglomeration AMG solver for linear elasticity in thin bodiesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 3 2009Prasad 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] Hybrid domain decomposition algorithms for compressible and almost incompressible elasticityINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2010Clark R. Dohrmann Abstract Overlapping Schwarz methods are considered for mixed finite element approximations of linear elasticity, with discontinuous pressure spaces, as well as for compressible elasticity approximated by standard conforming finite elements. The coarse components of the preconditioners are based on spaces, with a number of degrees of freedom per subdomain which are uniformly bounded, which are similar to those previously developed for scalar elliptic problems and domain decomposition methods of iterative substructuring type, i.e. methods based on nonoverlapping decompositions of the domain. The local components of the new preconditioners are based on solvers on a set of overlapping subdomains. In the current study, the dimension of the coarse spaces is smaller than in recently developed algorithms; in the compressible case all independent face degrees of freedom have been eliminated while in the almost incompressible case five out of six are not needed. In many cases, this will result in a reduction of the dimension of the coarse space by about one half compared with that of the algorithm previously considered. In addition, in spite of using overlapping subdomains to define the local components of the preconditioner, values of the residual and the approximate solution need only to be retained on the interface between the subdomains in the iteration of the new hybrid Schwarz algorithm. The use of discontinuous pressures makes it possible to work exclusively with symmetric, positive-definite problems and the standard preconditioned conjugate gradient method. Bounds are established for the condition number of the preconditioned operators. The bound for the almost incompressible case grows in proportion to the square of the logarithm of the number of degrees of freedom of individual subdomains and the third power of the relative overlap between the overlapping subdomains, and it is independent of the Poisson ratio as well as jumps in the Lamé parameters across the interface between the subdomains. Numerical results illustrate the findings. Copyright © 2009 John Wiley & Sons, Ltd. [source] A hybridizable discontinuous Galerkin method for linear elasticityINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2009S.-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] A corrected XFEM approximation without problems in blending elementsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2008Thomas-Peter Fries Abstract The extended finite element method (XFEM) enables local enrichments of approximation spaces. Standard finite elements are used in the major part of the domain and enriched elements are employed where special solution properties such as discontinuities and singularities shall be captured. In elements that blend the enriched areas with the rest of the domain problems arise in general. These blending elements often require a special treatment in order to avoid a decrease in the overall convergence rate. A modification of the XFEM approximation is proposed in this work. The enrichment functions are modified such that they are zero in the standard elements, unchanged in the elements with all their nodes being enriched, and varying continuously in the blending elements. All nodes in the blending elements are enriched. The modified enrichment function can be reproduced exactly everywhere in the domain and no problems arise in the blending elements. The corrected XFEM is applied to problems in linear elasticity and optimal convergence rates are achieved. Copyright © 2007 John Wiley & Sons, Ltd. [source] Robust and efficient domain decomposition preconditioners for adaptive hp finite element approximations of linear elasticity with and without discontinuous coefficientsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2004Andrew C. Bauer Abstract Adaptive finite element methods (FEM) generate linear equation systems that require dynamic and irregular patterns of storage, access, and computation, making their parallelization difficult. Additional difficulties are generated for problems in which the coefficients of the governing partial differential equations have large discontinuities. We describe in this paper the development of a set of iterative substructuring based solvers and domain decomposition preconditioners with an algebraic coarse-grid component that address these difficulties for adaptive hp approximations of linear elasticity with both homogeneous and inhomogeneous material properties. Our solvers are robust and efficient and place no restrictions on the mesh or partitioning. Copyright © 2003 John Wiley & Sons, Ltd. [source] Linear and non-linear finite element error estimation based on assumed strain fieldsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2002F. Gabaldón Abstract In this work we analyse the use of enhanced strain fields for the purpose of error estimation in finite element solid mechanics applications. The proposed approach evaluates the quality of the solution for standard Galerkin displacement elements, taking into account the enrichment of the solution with enhanced assumed strain mixed elements. The contribution of the enhanced strain modes is measured with an energy norm. The method proposed has two interesting advantages. Firstly, it results in a local formulation which is evaluated element by element. Secondly, it is easily extended to non-linear problems. In this work, the formulation is developed for linear elasticity, for finite strain elasticity, and von Mises small strain plasticity. Finally, some representative numerical simulations are presented which show in practice the performance of the method. Copyright © 2002 John Wiley & Sons, Ltd. [source] Efficient computation of order and mode of corner singularities in 3D-elasticityINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2001A. Dimitrov Abstract A general numerical procedure is presented for the efficient computation of corner singularities, which appear in the case of non-smooth domains in three-dimensional linear elasticity. 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 the problem is approximated by a Galerkin,Petrov finite element method. A quadratic eigenvalue problem (P+,Q+,2R) u=0 is obtained, with explicitly analytically defined matrices P,Q,R. Moreover, the three matrices are found to have optimal structure, so that P,R are symmetric and Q is skew symmetric, which can serve as an advantage in the following solution process. On this foundation a powerful iterative solution technique based on the Arnoldi method is submitted. For not too large systems this technique needs only one direct factorization of the banded matrix P for finding all eigenvalues in the interval ,e(,),(,0.5,1.0) (no eigenpairs can be ,lost') as well as the corresponding eigenvectors, which is a great improvement in comparison with the normally used determinant method. For large systems a variant of the algorithm with an incomplete factorization of P is implemented to avoid the appearance of too much fill-in. To illustrate the effectiveness of the present method several new numerical results are presented. In general, they show the dependence of the singular exponent on different geometrical parameters and the material properties. Copyright © 2001 John Wiley & Sons, Ltd. [source] A class of implicit variational inequalities and applications to frictional contactMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 14 2009Anca Capatina Abstract This paper deals with the mathematical and numerical analysis of a class of abstract implicit evolution variational inequalities. The results obtained here can be applied to a large variety of quasistatic contact problems in linear elasticity, including unilateral contact or normal compliance conditions with friction. In particular, a quasistatic unilateral contact problem with nonlocal friction is considered. An algorithm is derived and some numerical examples are presented. Copyright © 2009 John Wiley & Sons, Ltd. [source] Approximate boundary controllability for the system of linear elasticityMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 17 2004Andrzej Abstract The approximate controllability for variable coefficients, isotropic, evolution elasticity system is considered. The appropriate unique continuation theorem for solutions of the system is stated. Copyright © 2004 John Wiley & Sons, Ltd. [source] Homogenizing the acoustic properties of a porous matrix containing an incompressible inviscid fluidMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 10 2003J. L. Ferrin We undertake a rigorous derivation of the Biot's law for a porous elastic solid containing an inviscid fluid. We consider small displacements of a linear elastic solid being itself a connected periodic skeleton containing a pore structure of the characteristic size ,. It is completely saturated by an incompressible inviscid fluid. The model is described by the equations of the linear elasticity coupled with the linearized incompressible Euler system. We study the homogenization limit when the pore size ,tends to zero. The main difficulty is obtaining an a priori estimate for the gradient of the fluid velocity in the pore structure. Under the assumption that the solid part is connected and using results on the first order elliptic systems, we obtain the required estimate. It allows us to apply appropriate results from the 2-scale convergence. Then it is proved that the microscopic displacements and the fluid pressure converge in 2-scales towards a linear hyperbolic system for an effective displacement and an effective pressure field. Using correctors, we also give a strong convergence result. The obtained system is then compared with the Biot's law. It is found that there is a constitutive relation linking the effective pressure with the divergences of the effective fluid and solid displacements. Then we prove that the homogenized model coincides with the Biot's equations but with the added mass ,a being a matrix, which is calculated through an auxiliary problem in the periodic cell for the tortuosity. Furthermore, we get formulas for the matricial coefficients in the Biot's effective stress,strain relations. Finally, we consider the degenerate case when the fluid part is not connected and obtain Biot's model with the relative fluid displacement equal to zero. Copyright © 2003 John Wiley & Sons, Ltd. [source] A geometric-based algebraic multigrid method for higher-order finite element equations in two-dimensional linear elasticityNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 7 2009Yingxiong 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] A parallel block overlap preconditioning with inexact submatrix inversion for linear elasticity problemsNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 2 2002Igor 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] Polygonal finite element formulations for 2d linear-elastic FE problemsPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2009Markus Kraus Polygonal finite elements provide great flexibility meshing complex structures and for the refinement of meshes. The actual task in the development of these elements is to offer adequate and secure numerical results compared with regular finite elements at low computational costs. Particulary, finding an efficient and appropriate interpolation of the arbitrary element domain exhibits strong difficulties. Based on the general interpolant equation three element formulations are shown that use different interpolation strategies. The elements' performances are shown with a numerical example considering 2d linear elasticity. The results of the different element formulations are compared among each other, with analytical as well as with regular elements' results. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Mechanical Deformation of Compressible Chromatographic ColumnsBIOTECHNOLOGY PROGRESS, Issue 3 2002R. N. Keener A one-dimensional model of mechanical deformation of compressible chromatography columns is presented. The model is based on linear elasticity and continuum mechanics and is compared to a more complete two-dimensional model and one-dimensional porosity profiles measured by NMR imaging methods. The model provides a quantitative description of compression and the effects of wall support during scale-up. A simple criterion for the significance of wall support as a function of both diameter and length is also developed. Although the model accounts only for mechanical deformation, flow compression can be included, and validation presented here suggests that a more complete model may be valuable for anticipating the effects of scale and aspect ratio on pressure-flow behavior of compressible columns. [source] | |||||||||||||