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Solid Mechanics Problems (solid + mechanic_problem)
Selected AbstractsCertified solutions for hydraulic structures using the node-based smoothed point interpolation method (NS-PIM)INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 15 2010J. Cheng Abstract A meshfree node-based smoothed point interpolation method (NS-PIM), which has been recently developed for solid mechanics problems, is applied to obtain certified solutions with bounds for hydraulic structure designs. In this approach, shape functions for displacements are constructed using the point interpolation method (PIM), and the shape functions possess the Kronecker delta property and permit the straightforward enforcement of essential boundary conditions. The generalized smoothed Galerkin weak form is then applied to construct discretized system equations using the node-based smoothed strains. As a very novel and important property, the approach can obtain the upper bound solution in energy norm for hydraulic structures. A 2D gravity dam problem and a 3D arch dam problem are solved, respectively, using the NS-PIM and the simulation results of NS-PIM are found to be the upper bounds. Together with standard fully compatible FEM results as a lower bound, we have successfully determined the solution bounds to certify the accuracy of numerical solutions. This confirms that the NS-PIM is very useful for producing certified solutions for the analysis of huge hydraulic structures. Copyright © 2009 John Wiley & Sons, Ltd. [source] Upper and lower bounds for natural frequencies: A property of the smoothed finite element methodsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2010Zhi-Qian Zhang Abstract Node-based smoothed finite element method (NS-FEM) using triangular type of elements has been found capable to produce upper bound solutions (to the exact solutions) for force driving static solid mechanics problems due to its monotonic ,soft' behavior. This paper aims to formulate an NS-FEM for lower bounds of the natural frequencies for free vibration problems. To make the NS-FEM temporally stable, an ,-FEM is devised by combining the compatible and smoothed strain fields in a partition of unity fashion controlled by ,,[0, 1], so that both the properties of stiff FEM and the monotonically soft NS-FEM models can be properly combined for a desired purpose. For temporally stabilizing NS-FEM, , is chosen small so that it acts like a ,regularization parameter' making the NS-FEM stable, but still with sufficient softness ensuring lower bounds for natural frequency solution. Our numerical studies demonstrate that (1) using a proper ,, the spurious non-zero energy modes can be removed and the NS-FEM becomes temporally stable; (2) the stabilized NS-FEM becomes a general approach for solids to obtain lower bounds to the exact natural frequencies over the whole spectrum; (3) ,-FEM can even be tuned for obtaining nearly exact natural frequencies. Copyright © 2010 John Wiley & Sons, Ltd. [source] Decoupling and balancing of space and time errors in the material point method (MPM)INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2010Michael Steffen Abstract The material point method (MPM) is a computationally effective particle method with mathematical roots in both particle-in-cell and finite element-type methods. The method has proven to be extremely useful in solving solid mechanics problems involving large deformations and/or fragmentation of structures, problem domains that are sometimes problematic for finite element-type methods. Recently, the MPM community has focused significant attention on understanding the basic mathematical error properties of the method. Complementary to this thrust, in this paper we show how spatial and temporal errors are typically coupled within the MPM framework. In an attempt to overcome the challenge to analysis that this coupling poses, we take advantage of MPM's connection to finite element methods by developing a ,moving-mesh' variant of MPM that allows us to use finite element-type error analysis to demonstrate and understand the spatial and temporal error behaviors of MPM. We then provide an analysis and demonstration of various spatial and temporal errors in MPM and in simplified MPM-type simulations. Our analysis allows us to anticipate the global error behavior in MPM-type methods and allows us to estimate the time-step where spatial and temporal errors are balanced. Larger time-steps result in solutions dominated by temporal errors and show second-order temporal error convergence. Smaller time-steps result in solutions dominated by spatial errors, and hence temporal refinement produces no appreciative change in the solution. Based upon our understanding of MPM from both analysis and numerical experimentation, we are able to provide to MPM practitioners a collection of guidelines to be used in the selection of simulation parameters that respect the interplay between spatial (grid) resolution, number of particles and time-step. Copyright © 2009 John Wiley & Sons, Ltd. [source] A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part II applications to solid mechanics problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 9 2010G. R. Liu Abstract In part I of this paper, we have established the G space theory and fundamentals for W2 formulation. Part II focuses on the applications of the G space theory to formulate W2 models for solid mechanics problems. We first define a bilinear form, prove some of the important properties, and prove that the W2 formulation will be spatially stable, and convergent to exact solutions. We then present examples of some of the possible W2 models including the SFEM, NS-FEM, ES-FEM, NS-PIM, ES-PIM, and CS-PIM. We show the major properties of these models: (1) they are variationally consistent in a conventional sense, if the solution is sought in a proper H space (compatible cases); (2) They pass the standard patch test when the solution is sought in a proper G space with discontinuous functions (incompatible cases); (3) the stiffness of the discretized model is reduced compared with the finite element method (FEM) model and possibly to the exact model, allowing us to obtain upper bound solutions with respect to both the FEM and the exact solutions and (4) the W2 models are less sensitive to the quality of the mesh, and triangular meshes can be used without any accuracy problems. These properties and theories have been confirmed numerically via examples solved using a number of W2 models including compatible and incompatible cases. We shall see that the G space theory and the W2 forms can formulate a variety of stable and convergent numerical methods with the FEM as one special case. Copyright © 2009 John Wiley & Sons, Ltd. [source] Parametric enrichment adaptivity by the extended finite element methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2008Haim Waisman Abstract An adaptive method within the extended finite element method (XFEM) framework which adapts the enrichment function locally to the physics of a problem, as opposed to polynomial or mesh refinement, is presented. The method minimizes a local residual and determines the parameters of the enrichment function. We consider an energy form and a ,strong' form of the residual as error measures to drive the algorithm. Numerical examples for boundary layers and solid mechanics problems illustrate that the procedure converges. Moreover, when only the character of the solution is known, a good approximation is obtained in the area of interest. It is also shown that the method can be used to determine the order of singularities in solutions. Copyright © 2007 John Wiley & Sons, Ltd. [source] Adaptive superposition of finite element meshes in non-linear transient solid mechanics problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 9 2007Z. Yue Abstract An s-adaptive finite element procedure is developed for the transient analysis of 2-D solid mechanics problems with material non-linearity due to progressive damage. The resulting adaptive method simultaneously estimates and controls both the spatial error and temporal error within user-specified tolerances. The spatial error is quantified by the Zienkiewicz,Zhu error estimator and computed via superconvergent patch recovery, while the estimation of temporal error is based on the assumption of a linearly varying third-order time derivatives of the displacement field in conjunction with direct numerical time integration. The distinguishing characteristic of the s-adaptive procedure is the use of finite element mesh superposition (s-refinement) to provide spatial adaptivity. Mesh superposition proves to be particularly advantageous in computationally demanding non-linear transient problems since it is faster, simpler and more efficient than traditional h-refinement schemes. Numerical examples are provided to demonstrate the performance characteristics of the s-adaptive method for quasi-static and transient problems with material non-linearity. Copyright © 2007 John Wiley & Sons, Ltd. [source] A distributed memory parallel implementation of the multigrid method for solving three-dimensional implicit solid mechanics problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2004A. Namazifard Abstract We describe the parallel implementation of a multigrid method for unstructured finite element discretizations of solid mechanics problems. We focus on a distributed memory programming model and use the MPI library to perform the required interprocessor communications. We present an algebraic framework for our parallel computations, and describe an object-based programming methodology using Fortran90. The performance of the implementation is measured by solving both fixed- and scaled-size problems on three different parallel computers (an SGI Origin2000, an IBM SP2 and a Cray T3E). The code performs well in terms of speedup, parallel efficiency and scalability. However, the floating point performance is considerably below the peak values attributed to these machines. Lazy processors are documented on the Origin that produce reduced performance statistics. The solution of two problems on an SGI Origin2000, an IBM PowerPC SMP and a Linux cluster demonstrate that the algorithm performs well when applied to the unstructured meshes required for practical engineering analysis. Copyright © 2004 John Wiley & Sons, Ltd. [source] Meshfree point collocation method for elasticity and crack problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2004Sang-Ho Lee Abstract A generalized diffuse derivative approximation is combined with a point collocation scheme for solid mechanics problems. The derivatives are obtained from a local approximation so their evaluation is computationally very efficient. This meshfree point collocation method has other advantages: it does not require special treatment for essential boundary condition nor the time-consuming integration of a weak form. Neither the connectivity of the mesh nor differentiability of the weight function is necessary. The accuracy of the solutions is exceptional and generally exceeds that of element-free Galerkin method with linear basis. The performance and robustness are demonstrated by several numerical examples, including crack problems. Copyright © 2004 John Wiley & Sons, Ltd. [source] A vertex-based finite volume method applied to non-linear material problems in computational solid mechanicsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2003G. A. Taylor Abstract A vertex-based finite volume (FV) method is presented for the computational solution of quasi-static solid mechanics problems involving material non-linearity and infinitesimal strains. The problems are analysed numerically with fully unstructured meshes that consist of a variety of two- and three-dimensional element types. A detailed comparison between the vertex-based FV and the standard Galerkin FE methods is provided with regard to discretization, solution accuracy and computational efficiency. For some problem classes a direct equivalence of the two methods is demonstrated, both theoretically and numerically. However, for other problems some interesting advantages and disadvantages of the FV formulation over the Galerkin FE method are highlighted. Copyright © 2002 John Wiley & Sons, Ltd. [source] Meshfree weak,strong (MWS) form method and its application to incompressible flow problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10 2004G. R. Liu Abstract A meshfree weak,strong (MWS) form method has been proposed by the authors' group for linear solid mechanics problems based on a combined weak and strong form of governing equations. This paper formulates the MWS method for the incompressible Navier,Stokes equations that is non-linear in nature. In this method, the meshfree collocation method based on strong form equations is applied to the interior nodes and the nodes on the essential boundaries; the local Petrov,Galerkin weak form is applied only to the nodes on the natural boundaries of the problem domain. The MWS method is then applied to simulate the steady problem of natural convection in an enclosed domain and the unsteady problem of viscous flow around a circular cylinder using both regular and irregular nodal distributions. The simulation results are validated by comparing with those of other numerical methods as well as experimental data. It is demonstrated that the MWS method has very good efficiency and accuracy for fluid flow problems. It works perfectly well for irregular nodes using only local quadrature cells for nodes on the natural boundary, which can be generated without any difficulty. Copyright © 2004 John Wiley & Sons, Ltd. [source] | ||||||||||