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Weak Form (weak + form)

Distribution by Scientific Domains
Engineering88%
Business, Economics, Finance and Accounting7%

Kinds of Weak Form

  • galerkin weak form
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    Selected Abstracts

    Torsional balance of plan-asymmetric structures with frictional dampers: experimental results

    EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, Issue 15 2006
    Ignacio J. Vial
    Abstract This investigation deals with the measured seismic response of a six-storey asymmetric structural model with frictional dampers. Its main objective is to experimentally prove the concept of weak torsional balance for mass- and stiffness-eccentric model configurations. The goal is to control the torsional response of these asymmetric structures and to achieve, if possible, a weak form of torsional balance by placing the so-called empirical centre of balance (ECB) of the structure at equal distance from the edges of the building plan. The control of the dynamic response of asymmetric structures is investigated herein by using steel,teflon frictional dampers. As expected from theory, experimental results show that the mean-square and peak displacement demand at the flexible and stiff edges of the plan may be similar in magnitude if the dampers are optimally placed. Frictional dampers have proven equally effective in controlling lateral-torsional coupling of torsionally flexible as well as stiff structures. On the other hand, it is shown that impulsive ground motions require larger frictional capacities to achieve weak torsional balance. Copyright © 2006 John Wiley & Sons, Ltd. [source]

    Some numerical issues using element-free Galerkin mesh-less method for coupled hydro-mechanical problems

    INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 7 2009
    Mohammad Norouz Oliaei
    Abstract A new formulation of the element-free Galerkin (EFG) method is developed for solving coupled hydro-mechanical problems. The numerical approach is based on solving the two governing partial differential equations of equilibrium and continuity of pore water simultaneously. Spatial variables in the weak form, i.e. displacement increment and pore water pressure increment, are discretized using the same EFG shape functions. An incremental constrained Galerkin weak form is used to create the discrete system equations and a fully implicit scheme is used for discretization in the time domain. Implementation of essential boundary conditions is based on a penalty method. Numerical stability of the developed formulation is examined in order to achieve appropriate accuracy of the EFG solution for coupled hydro-mechanical problems. Examples are studied and compared with closed-form or finite element method solutions to demonstrate the validity of the developed model and its capabilities. The results indicate that the EFG method is capable of handling coupled problems in saturated porous media and can predict well both the soil deformation and variation of pore water pressure over time. Some guidelines are proposed to guarantee the accuracy of the EFG solution for coupled hydro-mechanical problems. Copyright © 2008 John Wiley & Sons, Ltd. [source]

    Conserving Galerkin weak formulations for computational fracture mechanics

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 12 2002
    Shaofan Li
    Abstract In this paper, a notion of invariant Galerkin-variational weak forms is proposed. Two specific invariant variational weak forms, the J-invariant and the L-invariant, are constructed based on the corresponding conservation laws in elasticity, one of which is the conservation of Eshelby's energy-momentum (Eshelby. Philos. Trans. Roy. Soc. 1951; 87: 12; In Solid State Physics, Setitz F, Turnbull D (eds). Academic Press: New York, 1956; 331; Rice, J. Appl. Mech. 1968; 35: 379). It is shown that the finite element solution obtained from the invariant Galerkin weak formulations proposed here can conserve the value of J-integral, or L-integral exactly. In other words, the J and L integrals of the Galerkin finite element solutions are path independent in the discrete sense. It is argued that by using the J-invariant Galerkin weak form to compute near crack-tip field in an elastic solid, one may accurately calculate the crack extension energy release rate and subsequently the stress intensity factors in numerical computations, because the flux of the energy-momentum is conserved in discrete computations. This may provide an alternative means to accurately simulate crack growth and propagation. Copyright © 2002 John Wiley & Sons, Ltd. [source]

    Higher-order XFEM for curved strong and weak discontinuities

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2010
    Kwok Wah Cheng
    Abstract The extended finite element method (XFEM) enables the accurate approximation of solutions with jumps or kinks within elements. Optimal convergence rates have frequently been achieved for linear elements and piecewise planar interfaces. Higher-order convergence for arbitrary curved interfaces relies on two major issues: (i) an accurate quadrature of the Galerkin weak form for the cut elements and (ii) a careful formulation of the enrichment, which should preclude any problems in the blending elements. For (i), we employ a strategy of subdividing the elements into subcells with only one curved side. Reference elements that are higher-order on only one side are then used to map the integration points to the real element. For (ii), we find that enrichments for strong discontinuities are easily extended to higher-order accuracy. In contrast, problems in blending elements may hinder optimal convergence for weak discontinuities. Different formulations are investigated, including the corrected XFEM. Numerical results for several test cases involving strong or weak curved discontinuities are presented. Quadratic and cubic approximations are investigated. Optimal convergence rates are achieved using the standard XFEM for the case of a strong discontinuity. Close-to-optimal convergence rates for the case of a weak discontinuity are achieved using the corrected XFEM. 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 I theory

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 9 2010
    G. R. Liu
    Abstract This paper introduces a G space theory and a weakened weak form (W2) using the generalized gradient smoothing technique for a unified formulation of a wide class of compatible and incompatible methods. The W2 formulation works for both finite element method settings and mesh-free settings, and W2 models can have special properties including softened behavior, upper bounds and ultra accuracy. Part I of this paper focuses on the theory and fundamentals for W2 formulations. A normed G space is first defined to include both continuous and discontinuous functions allowing the use of much more types of methods/techniques to create shape functions for numerical models. Important properties and a set of useful inequalities for G spaces are then proven in the theory and analyzed in detail. These properties ensure that a numerical method developed based on the W2 formulation will be spatially stable and convergent to the exact solutions, as long as the physical problem is well posed. The theory is applicable to any problems to which the standard weak formulation is applicable, and can offer numerical solutions with special properties including ,close-to-exact' stiffness, upper bounds and ultra accuracy. Copyright © 2009 John Wiley & Sons, Ltd. [source]

    A finite element formulation for thermoelastic damping analysis

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 6 2009
    Enrico Serra
    Abstract We present a finite element formulation based on a weak form of the boundary value problem for fully coupled thermoelasticity. The thermoelastic damping is calculated from the irreversible flow of entropy due to the thermal fluxes that have originated from the volumetric strain variations. Within our weak formulation we define a dissipation function that can be integrated over an oscillation period to evaluate the thermoelastic damping. We show the physical meaning of this dissipation function in the framework of the well-known Biot's variational principle of thermoelasticity. The coupled finite element equations are derived by considering harmonic small variations of displacement and temperature with respect to the thermodynamic equilibrium state. In the finite element formulation two elements are considered: the first is a new 8-node thermoelastic element based on the Reissner,Mindlin plate theory, which can be used for modeling thin or moderately thick structures, while the second is a standard three-dimensional 20-node iso-parametric thermoelastic element, which is suitable to model massive structures. For the 8-node element the dissipation along the plate thickness has been taken into account by introducing a through-the-thickness dependence of the temperature shape function. With this assumption the unknowns and the computational effort are minimized. Comparisons with analytical results for thin beams are shown to illustrate the performances of those coupled-field elements. Copyright © 2008 John Wiley & Sons, Ltd. [source]

    A Hermite reproducing kernel approximation for thin-plate analysis with sub-domain stabilized conforming integration

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2008
    Dongdong Wang
    Abstract A Hermite reproducing kernel (RK) approximation and a sub-domain stabilized conforming integration (SSCI) are proposed for solving thin-plate problems in which second-order differentiation is involved in the weak form. Although the standard RK approximation can be constructed with an arbitrary order of continuity, the proposed approximation based on both deflection and rotation variables is shown to be more effective in solving plate problems. By imposing the Kirchhoff mode reproducing conditions on deflectional and rotational degrees of freedom simultaneously, it is demonstrated that the minimum normalized support size (coverage) of kernel functions can be significantly reduced. With this proposed approximation, the Galerkin meshfree framework for thin plates is then formulated and the integration constraint for bending exactness is also derived. Subsequently, an SSCI method is developed to achieve the exact pure bending solution as well as to maintain spatial stability. Numerical examples demonstrate that the proposed formulation offers superior convergence rates, accuracy and efficiency, compared with those based on higher-order Gauss quadrature rule. Copyright © 2007 John Wiley & Sons, Ltd. [source]

    The immersed/fictitious element method for fluid,structure interaction: Volumetric consistency, compressibility and thin members

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2008
    Hongwu Wang
    Abstract A weak form and an implementation are given for fluid,structure interaction by the immersed/fictitious element method for compressible fluids. The weak form is applicable to models where the fluid is described by Eulerian coordinates while the solid is described by Lagrangian coordinates, which suits their intrinsic characteristics. A unique feature of the method is the treatment of the fictitious fluid by a Lagrangian description, which simplifies the interface conditions. Methods for enforcing volumetric consistency between the fluid and solid and treating thin members are given. Although a compressible viscous fluid is considered here, the new developments can be applied to incompressible fluids. Copyright © 2007 John Wiley & Sons, Ltd. [source]

    Wavelet Galerkin method in multi-scale homogenization of heterogeneous media

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2006
    Shafigh Mehraeen
    Abstract The hierarchical properties of scaling functions and wavelets can be utilized as effective means for multi-scale homogenization of heterogeneous materials under Galerkin framework. It is shown in this work, however, when the scaling functions are used as the shape functions in the multi-scale wavelet Galerkin approximation, the linear dependency in the scaling functions renders improper zero energy modes in the discrete differential operator (stiffness matrix) if integration by parts is invoked in the Galerkin weak form. An effort is made to obtain the analytical expression of the improper zero energy modes in the wavelet Galerkin differential operator, and the improper nullity of the discrete differential operator is then removed by an eigenvalue shifting approach. A unique property of multi-scale wavelet Galerkin approximation is that the discrete differential operator at any scale can be effectively obtained. This property is particularly useful in problems where the multi-scale solution cannot be obtained simply by a wavelet projection of the finest scale solution without utilizing the multi-scale discrete differential operator, for example, the multi-scale analysis of an eigenvalue problem with oscillating coefficients. Copyright © 2005 John Wiley & Sons, Ltd. [source]

    Strong and weak arbitrary discontinuities in spectral finite elements

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2005
    A. Legay
    Abstract Methods for constructing arbitrary discontinuities within spectral finite elements are described and studied. We use the concept of the eXtended Finite Element Method (XFEM), which introduces the discontinuity through a local partition of unity, so there is no requirement for the mesh to be aligned with the discontinuities. A key aspect of the implementation of this method is the treatment of the blending elements adjacent to the local partition of unity. We found that a partition constructed from spectral functions one order lower than the continuous approximation is optimal and no special treatment is needed for higher order elements. For the quadrature of the Galerkin weak form, since the integrand is discontinuous, we use a strategy of subdividing the discontinuous elements into 6- and 10-node triangles; the order of the element depends on the order of the spectral method for curved discontinuities. Several numerical examples are solved to examine the accuracy of the methods. For straight discontinuities, we achieved the optimal convergence rate of the spectral element. For the curved discontinuity, the convergence rate in the energy norm error is suboptimal. We attribute the suboptimality to the approximations in the quadrature scheme. We also found that modification of the adjacent elements is only needed for lower order spectral elements. Copyright © 2005 John Wiley & Sons, Ltd. [source]

    Arbitrary discontinuities in space,time finite elements by level sets and X-FEM

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 15 2004
    Jack Chessa
    Abstract An enriched finite element method with arbitrary discontinuities in space,time is presented. The discontinuities are treated by the extended finite element method (X-FEM), which uses a local partition of unity enrichment to introduce discontinuities along a moving hyper-surface which is described by level sets. A space,time weak form for conservation laws is developed where the Rankine,Hugoniot jump conditions are natural conditions of the weak form. The method is illustrated in the solution of first order hyperbolic equations and applied to linear first order wave and non-linear Burgers' equations. By capturing the discontinuity in time as well as space, results are improved over capturing the discontinuity in space alone and the method is remarkably accurate. Implications to standard semi-discretization X-FEM formulations are also discussed. Copyright © 2004 John Wiley & Sons, Ltd. [source]

    Meshfree point collocation method for elasticity and crack problems

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2004
    Sang-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]

    Stabilized finite element method for viscoplastic flow: formulation with state variable evolution

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2003
    Antoinette M. Maniatty
    Abstract A stabilized, mixed finite element formulation for modelling viscoplastic flow, which can be used to model approximately steady-state metal-forming processes, is presented. The mixed formulation is expressed in terms of the velocity, pressure and state variable fields, where the state variable is used to describe the evolution of the material's resistance to plastic flow. The resulting system of equations has two sources of well-known instabilities, one due to the incompressibility constraint and one due to the convection-type state variable equation. Both of these instabilities are handled by adding mesh-dependent stabilization terms, which are functions of the Euler,Lagrange equations, to the usual Galerkin method. Linearization of the weak form is derived to enable a Newton,Raphson implementation into an object-oriented finite element framework. A progressive solution strategy is used for improving convergence for highly non-linear material behaviour, typical for metals. Numerical experiments using the stabilization method with hierarchic shape functions for the velocity, pressure and state variable fields in viscoplastic flow and metal-forming problems show that the stabilized finite element method is effective and efficient for non-linear steady forming problems. Finally, the results are discussed and conclusions are inferred. Copyright © 2002 John Wiley & Sons, Ltd. [source]

    The extended finite element method for rigid particles in Stokes flow

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2001
    G. J. Wagner
    Abstract A new method for the simulation of particulate flows, based on the extended finite element method (X-FEM), is described. In this method, the particle surfaces need not conform to the finite element boundaries, so that moving particles can be simulated without remeshing. The near field form of the fluid flow about each particle is built into the finite element basis using a partition of unity enrichment, allowing the simple enforcement of boundary conditions and improved accuracy over other methods on a coarse mesh. We present a weak form of the equations of motion useful for the simulation of freely moving particles, and solve example problems for particles with prescribed and unknown velocities. Copyright © 2001 John Wiley & Sons, Ltd. [source]

    Variance-reduced Monte Carlo solutions of the Boltzmann equation for low-speed gas flows: A discontinuous Galerkin formulation

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2008
    Lowell L. Baker
    Abstract We present and discuss an efficient, high-order numerical solution method for solving the Boltzmann equation for low-speed dilute gas flows. The method's major ingredient is a new Monte Carlo technique for evaluating the weak form of the collision integral necessary for the discontinuous Galerkin formulation used here. The Monte Carlo technique extends the variance reduction ideas first presented in Baker and Hadjiconstantinou (Phys. Fluids 2005; 17, art. no. 051703) and makes evaluation of the weak form of the collision integral not only tractable but also very efficient. The variance reduction, achieved by evaluating only the deviation from equilibrium, results in very low statistical uncertainty and the ability to capture arbitrarily small deviations from equilibrium (e.g. low-flow speed) at a computational cost that is independent of the magnitude of this deviation. As a result, for low-signal flows the proposed method holds a significant computational advantage compared with traditional particle methods such as direct simulation Monte Carlo (DSMC). Copyright © 2008 John Wiley & Sons, Ltd. [source]

    Meshfree weak,strong (MWS) form method and its application to incompressible flow problems

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10 2004
    G. 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]

    Linear Programming and the von Neumann Model

    METROECONOMICA, Issue 1 2000
    Christian Bidard
    The formal similarity between von Neumann's theorem on maximal growth and (a weak form of) the fundamental theorem of linear programming is striking. The parallelism is explained by considering a simple economy for which the two problems are identical. [source]

    An adaptive wavelet viscosity method for hyperbolic conservation laws

    NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2008
    Daniel Castaño Díez
    Abstract We extend the multiscale finite element viscosity method for hyperbolic conservation laws developed in terms of hierarchical finite element bases to a (pre-orthogonal spline-)wavelet basis. Depending on an appropriate error criterion, the multiscale framework allows for a controlled adaptive resolution of discontinuities of the solution. The nonlinearity in the weak form is treated by solving a least-squares data fitting problem. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008 [source]

    Coupling Techniques for Thermal and Mechanical Fluid-Structure-Interactions in Aeronautics

    PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2005
    Matthias Haupt
    For the coupled thermal and mechanical analysis of spacecraft structures a simulation environment was developed containing the necessary coupling techniques. The numerical concept uses the weak form of the interface conditions on the coupling surface. The iterative solution of the coupled equations is based on the classical Dirichlet-Neumann approach. Transient problems are handled with iterative staggered schemes. A flexible component-based software environment combines existing fluid and structural analysis codes. Aspects of the architecture and its implementation are described. Finally an application to a spacecraft structure is shown. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

    3D Surface Smoothing for Arbitrary FE-Meshes by Means of an Enhanced Surface Description

    PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2003
    P. Helnwein Dipl.-Ing.
    This paper describes a novel approach to the problem of 3D surface smoothing by introducing an enhanced surface description of the structural model. The idea of the enhanced surface model is the introduction of an independent vector field for the description of the surface normal field. It is connected to the discretized surface by means of a weak form of the orthogonality conditions. Surface smoothing then is applied element-by-element using the interpolation of the surface patch and the nodal normal vectors. [source]

    MEAN REVERSION OF THE FISCAL CONDUCT IN 24 DEVELOPING COUNTRIES

    THE MANCHESTER SCHOOL, Issue 4 2010
    AHMAD ZUBAIDI BAHARUMSHAH
    In this paper, we examine the mean reverting behaviour of fiscal deficit by analysing the fiscal position of 24 developing countries. Using annual data over the period 1970,2003 and the series-specific panel unit root test developed by Breuer et al. (Oxford Bulletin of Economics and Statistics, Vol. 64 (2002), pp. 527,546), we found the budget process for most developing countries fails to satisfy the strong-form sustainability condition. Further investigation shows the budget process for a majority of the countries is on a sustainable path (weak form) when a one-time, structural break is allowed in the model. Therefore, our empirical results suggest that the budget process in most of the sample countries is in accordance with the intertemporal budget constraint. [source]

    A new mixed finite element method for poro-elasticity

    INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 6 2008
    Maria Tchonkova
    Abstract Development of robust numerical solutions for poro-elasticity is an important and timely issue in modern computational geomechanics. Recently, research in this area has seen a surge in activity, not only because of increased interest in coupled problems relevant to the petroleum industry, but also due to emerging applications of poro-elasticity for modelling problems in biomedical engineering and materials science. In this paper, an original mixed least-squares method for solving Biot consolidation problems is developed. The solution is obtained via minimization of a least-squares functional, based upon the equations of equilibrium, the equations of continuity and weak forms of the constitutive relationships for elasticity and Darcy flow. The formulation involves four separate categories of unknowns: displacements, stresses, fluid pressures and velocities. Each of these unknowns is approximated by linear continuous functions. The mathematical formulation is implemented in an original computer program, written from scratch and using object-oriented logic. The performance of the method is tested on one- and two-dimensional classical problems in poro-elasticity. The numerical experiments suggest the same rates of convergence for all four types of variables, when the same interpolation spaces are used. The continuous linear triangles show the same rates of convergence for both compressible and entirely incompressible elastic solids. This mixed formulation results in non-oscillating fluid pressures over entire domain for different moments of time. The method appears to be naturally stable, without any need of additional stabilization terms with mesh-dependent parameters. Copyright © 2007 John Wiley & Sons, Ltd. [source]

    A variational multiscale model for the advection,diffusion,reaction equation

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 7 2009
    Guillaume Houzeaux
    Abstract The variational multiscale (VMS) method sets a general framework for stabilization methods. By splitting the exact solution into coarse (grid) and fine (subgrid) scales, one can obtain a system of two equations for these unknowns. The grid scale equation is solved using the Galerkin method and contains an additional term involving the subgrid scale. At this stage, several options are usually considered to deal with the subgrid scale equation: this includes the choice of the space where the subgrid scale would be defined as well as the simplifications leading to compute the subgrid scale analytically or numerically. The present study proposes to develop a two-scale variational method for the advection,diffusion,reaction equation. On the one hand, a family of weak forms are obtained by integrating by parts a fraction of the advection term. On the other hand, the solution of the subgrid scale equation is found using the following. First, a two-scale variational method is applied to the one-dimensional problem. Then, a series of approximations are assumed to solve the subgrid space equation analytically. This allows to devise expressions for the ,stabilization parameter' ,, in the context of VMS (two-scale) method. The proposed method is equivalent to the traditional Green's method used in the literature to solve residual-free bubbles, although it offers another point of view, as the strong form of the subgrid scale equation is solved explicitly. In addition, the authors apply the methodology to high-order elements, namely quadratic and cubic elements. The proposed model consists in assuming that the subgrid scale vanishes also on interior nodes of the element and applying the strategy used for linear element in the segment between these interior nodes. The proposed scheme is compared with existing ones through the solution of a one-dimensional numerical example for linear, quadratic and cubic elements. In addition, the mesh convergence is checked for high-order elements through the solution of an exact solution in two dimensions. Copyright © 2008 John Wiley & Sons, Ltd. [source]

    Conserving Galerkin weak formulations for computational fracture mechanics

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 12 2002
    Shaofan Li
    Abstract In this paper, a notion of invariant Galerkin-variational weak forms is proposed. Two specific invariant variational weak forms, the J-invariant and the L-invariant, are constructed based on the corresponding conservation laws in elasticity, one of which is the conservation of Eshelby's energy-momentum (Eshelby. Philos. Trans. Roy. Soc. 1951; 87: 12; In Solid State Physics, Setitz F, Turnbull D (eds). Academic Press: New York, 1956; 331; Rice, J. Appl. Mech. 1968; 35: 379). It is shown that the finite element solution obtained from the invariant Galerkin weak formulations proposed here can conserve the value of J-integral, or L-integral exactly. In other words, the J and L integrals of the Galerkin finite element solutions are path independent in the discrete sense. It is argued that by using the J-invariant Galerkin weak form to compute near crack-tip field in an elastic solid, one may accurately calculate the crack extension energy release rate and subsequently the stress intensity factors in numerical computations, because the flux of the energy-momentum is conserved in discrete computations. This may provide an alternative means to accurately simulate crack growth and propagation. Copyright © 2002 John Wiley & Sons, Ltd. [source]

    Meshfree simulation of failure modes in thin cylinders subjected to combined loads of internal pressure and localized heat

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2008
    Dong Qian
    Abstract This paper focuses on the non-linear responses in thin cylindrical structures subjected to combined mechanical and thermal loads. The coupling effects of mechanical deformation and temperature in the material are considered through the development of a thermo-elasto-viscoplastic constitutive model at finite strain. A meshfree Galerkin approach is used to discretize the weak forms of the energy and momentum equations. Due to the different time scales involved in thermal conduction and failure development, an explicit,implicit time integration scheme is developed to link the time scale differences between the two key mechanisms. We apply the developed approach to the analysis of the failure of cylindrical shell subjected to both heat sources and internal pressure. The numerical results show four different failure modes: dynamic fragmentation, single crack with branch, thermally induced cracks and cracks due to the combined effects of pressure and temperature. These results illustrate the important roles of thermal and mechanical loads with different time scales. Copyright © 2008 John Wiley & Sons, Ltd. [source]