Displacement/pressure mixed interpolation in the method of finite spheres

The displacement‐based formulation of the method of finite spheres is observed to exhibit volumetric ‘locking’ when incompressible or nearly incompressible deformations are encountered. In this paper, we present a displacement/pressure mixed formulation as a solution to this problem. We analyse the stability and optimality of the formulation for several discretization schemes using numerical inf–sup tests. Issues concerning computational efficiency are also discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

An explicit discontinuous Galerkin method for non‐linear solid dynamics: Formulation,parallel implementation and scalability properties

An explicit‐dynamics spatially discontinuous Galerkin (DG) formulation for non‐linear solid dynamics is proposed and implemented for parallel computation. DG methods have particular appeal in problems involving complex material response, e.g. non‐local behavior and failure, as, even in the presence of discontinuities, they provide a rigorous means of ensuring both consistency and stability. In the proposed method, these are guaranteed: the former by the use of average numerical fluxes and the latter by the introduction of appropriate quadratic terms in the weak formulation. The semi‐discrete system of ordinary differential equations is integrated in time using a conventional second‐order central‐difference explicit scheme. A stability criterion for the time integration algorithm, accounting for the influence of the DG discretization stability, is derived for the equivalent linearized system. This approach naturally lends itself to efficient parallel implementation. The resulting DG computational framework is implemented in three dimensions via specialized interface elements. The versatility, robustness and scalability of the overall computational approach are all demonstrated in problems involving stress‐wave propagation and large plastic deformations. Copyright © 2007 John Wiley & Sons, Ltd.     

A two-level mesh repartitioning scheme for the displacement-based lower-order finite element methods in volumetric locking-free analyses

We present an approach for repartitioning existing lower-order finite element mesh based on quadrilateral or triangular elements for the linear and nonlinear volumetric locking-free analysis. This approach contains two levels of mesh repartitioning. The first-level mesh re-partitioning is an h-adaptive mesh refinement for the generation of a refined mesh needed in the second-level mesh coarsening. The second-level mesh coarsening involves a gradient smoothing scheme performed on each pair of adjacent elements selected based on the first-level refined mesh. With the repartitioned mesh and smoothed gradient, the equivalence between the mixed finite element formulation and the displacement-based finite element formulation is established. The extension to nonlinear finite element formulation is also considered. Several linear and non-linear numerical benchmarks are solved and numerical inf-sup tests are conducted to demonstrate the accuracy and stability of the proposed formulation in the nearly incompressible applications.     

A large deformation frictional contact formulation using NURBS‐based isogeometric analysis

This paper focuses on the application of NURBS‐based isogeometric analysis to Coulomb frictional contact problems between deformable bodies, in the context of large deformations. A mortar‐based approach is presented to treat the contact constraints, whereby the discretization of the continuum is performed with arbitrary order NURBS, as well as C⁰‐continuous Lagrange polynomial elements for comparison purposes. The numerical examples show that the proposed contact formulation in conjunction with the NURBS discretization delivers accurate and robust predictions. Results of lower quality are obtained from the Lagrange discretization, as well as from a different contact formulation based on the enforcement of the contact constraints at every integration point on the contact surface. Copyright © 2011 John Wiley & Sons, Ltd.     

On the evaluation of the method of finite spheres for the solution of Reissner–Mindlin plate problems using the numerical inf–sup test

In this paper we use the numerical inf–sup test to evaluate both displacement‐based and mixed discretization schemes for the solution of Reissner–Mindlin plate problems using the meshfree method of finite spheres. While an analytical proof of whether a discretization scheme passes the inf–sup condition is most desirable, such a proof is usually out of reach due to the complexity of the meshfree approximation spaces involved. The numerical inf–sup test (Int. J. Numer. Meth. Engng 1997; 40 :3639–3663), developed to test finite element discretization spaces, has therefore been adopted in this paper. Tests have been performed for both regular and irregular nodal configurations. While, like linear finite elements, pure displacement‐based approximation spaces with linear consistency do not pass the inf–sup test and exhibit shear locking, quadratic discretizations, unlike quadratic finite elements, pass the test. Pure displacement‐based and mixed approximation spaces that pass the numerical inf–sup test exhibit optimal or near optimal convergence behaviour. Copyright © 2007 John Wiley & Sons, Ltd.     

Symmetry preserving algorithm for large displacement frictionless contact by the pre‐discretization penalty method

A three‐dimensional contact algorithm based on the pre‐discretization penalty method is presented. Using the pre‐discretization formulation gives rise to contact searching performed at the surface Gaussian integration points. It is shown that the proposed method is consistent with the continuum formulation of the problem and allows an easy incorporation of higher‐order elements with midside nodes to the analysis. Moreover, a symmetric treatment of mutually contacting surfaces is preserved even under large displacement increments. The proposed algorithm utilizes the BFGS method modified for constrained non‐linear systems. The effectiveness of quadratic isoparametric elements in contact analysis is tested in terms of numerical examples verified by analytical solutions and experimental measurements. The symmetry of the algorithm is clearly manifested in the problem of impact of two elastic cylinders. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical and spectral investigations of Trefftz infinite elements

The numerical and spectral performance of novel infinite elements for exterior problems of time‐harmonic acoustics are examined. The formulation is based on a functional which provides a general framework for domain‐based computation of exterior problems. Two prominent features simplify the task of discretization: the infinite elements mesh the interface only and need not match the finite elements on the interface. Various infinite element approximations for two‐dimensional configurations with circular interfaces are reviewed. Numerical results demonstrate the good performance of these schemes. A simple study points to the proper interpretation of spectral results for the formulation. The spectral properties of these infinite elements are examined with a view to the representation of physics and efficient numerical solution. Copyright © 1999 John Wiley & Sons, Ltd.     

On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far‐field boundary treatment

A reduced order model (ROM) based on the proper orthogonal decomposition (POD)/Galerkin projection method is proposed as an alternative discretization of the linearized compressible Euler equations. It is shown that the numerical stability of the ROM is intimately tied to the choice of inner product used to define the Galerkin projection. For the linearized compressible Euler equations, a symmetry transformation motivates the construction of a weighted L² inner product that guarantees certain stability bounds satisfied by the ROM. Sufficient conditions for well‐posedness and stability of the present Galerkin projection method applied to a general linear hyperbolic initial boundary value problem (IBVP) are stated and proven. Well‐posed and stable far‐field and solid wall boundary conditions are formulated for the linearized compressible Euler ROM using these more general results. A convergence analysis employing a stable penalty‐like formulation of the boundary conditions reveals that the ROM solution converges to the exact solution with refinement of both the numerical solution used to generate the ROM and of the POD basis. An a priori error estimate for the computed ROM solution is derived, and examined using a numerical test case. Published in 2010 by John Wiley & Sons, Ltd.     

Incompressibility in the multimodel Arlequin framework

In this paper, we show how one can account for the incompressibility constraint within the multimodel Arlequin framework. The main issue is the treatment of a double constraining of the displacement fields in the gluing model zone. An elastic incompressible structure problem is considered. An incompressible Arlequin formulation of this problem is developed and supported by a stability and consistency result. Its discretization by means of mixed finite elements is detailed. The Inf–Sup condition is numerically discussed for different choices of the Arlequin method parameters. Several numerical tests are conducted and practical recommendations for appropriate choices of these parameters are clearly stated. Copyright © 2013 John Wiley & Sons, Ltd.     

A new mass lumping scheme for the mixed hybrid finite element method

Diffusion‐type partial differential equation is a common mathematical model in physics. Solved by mixed finite elements, it leads to a system matrix which is not always an M‐matrix. Therefore, the numerical solution may exhibit unphysical results due to oscillations. The criterion necessary to obtain an M‐matrix is discussed in details for triangular, rectangular and tetrahedral elements. It is shown that the system matrix is never an M‐matrix for rectangular elements and can be an M‐matrix for triangular an tetrahedral elements if criteria on the element's shape and on the time step length are fulfilled. A new mass lumping scheme is developed which leads to a less restrictive criterion: the discretization must be weakly acute (all angles less than π/2) and there is no constraint on the time step length. The lumped formulation of mixed hybrid finite element can be applied not only to triangular meshes but also to more general shape elements in two and three dimensions. Numerical experiments show that, compared to the standard mixed hybrid formulation, the lumping scheme avoids (or strongly reduce) oscillations and does not create additional numerical errors. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical manifold method for dynamic consolidation of saturated porous media with three-field formulation

In terms of the three-field formulation of Biot's dynamic consolidation theory, the numerical manifold method (NMM) is developed, where the same approximation to skeleton displacement ( u ) and fluid velocity ( w ) is employed and able to reflect incompressible as well as compressible deformation, but the approximation to pore pressure (p ) takes two different types, respectively. The first type of approximation to p is continuous piecewise linear interpolation and the second type assumes that p is a constant within each element. It is verified that using the second type of approximation to p naturally satisfies the inf-sup condition even in the limits of rigid skeleton and very low permeability, avoiding the locking problem accordingly. Energy components done by various forces are calculated to verify the accuracy and stability of the time integration scheme. A mass lumping technique in the NMM framework is employed to effectively reduce the unphysical oscillations and increase computational efficiency, which is another unique advantage of NMM over other numerical methods. A number of numerical tests are conducted to demonstrate the robustness and versatility of the proposed mixed NMM models.     

On non-linear dynamics of shells: implementation of energy-momentum conserving algorithm for a finite rotation shell model

Continuum and numerical formulations for non-linear dynamics of thin shells are presented in this work. An elastodynamic shell model is developed from the three-dimensional continuum by employing standard assumptions of the first-order shear-deformation theories. Motion of the shell-director is described by a singularity-free formulation based on the rotation vector. Temporal discretization is performed by an implicit, one-step, second-order accurate, time-integration scheme. In this work, an energy and momentum conserving algorithm, which exactly preserves the fundamental constants of the shell motion and guaranties unconditional algorithmic stability, is used. It may be regarded as a modification of the standard mid-point rule. Spatial discretization is based on the four-noded isoparametric element. Particular attention is devoted to the consistent linearization of the weak form of the initial boundary value problem discretized in time and space, in order to achieve a quadratic rate of asymptotic convergence typical for the Newton–Raphson based solution procedures. An unconditionally stable time finite element formulation suitable for the long-term dynamic computations of flexible shell-like structures, which may be undergoing large displacements, large rotations and large motions is therefore obtained. A set of numerical examples is presented to illustrate the present approach and the performance of the isoparametric four-noded shell finite element in conjunction with the implicit energy and momentum conserving time-integration algorithm. © 1998 John Wiley & Sons, Ltd.     

A radial point interpolation meshless method extended with an elastic rate-independent continuum damage model for concrete materials

An advanced discretization meshless technique, the radial point interpolation method (RPIM), is applied to analyze concrete structures using an elastic continuum damage constitutive model. Here, the theoretical basis of the material model and the computational procedure are fully presented. The plane stress meshless formulation is extended to a rate-independent damage criterion, where both compressive and tensile damage evolutions are established based on a Helmholtz free energy function. Within the return-mapping damage algorithm, the required variable fields, such as the damage variables and the displacement field, are obtained. This study uses the Newton–Raphson nonlinear solution algorithm to achieve the nonlinear damage solution. The verification, where the performance is assessed, of the proposed model is demonstrated by relevant numerical examples available in the literature.     

Spectral element methods for nonlinear spatio-temporal dynamics of an Euler-Bernoulli beam

Spectral element methods are high order accurate methods which have been successfully utilized for solving ordinary and partial differential equations. In this paper the space-time spectral element (STSE) method is employed to solve a simply supported modified Euler-Bernoulli nonlinear beam undergoing forced lateral vibrations. This system was chosen for analysis due to the availability of a reference solution of the form of a forced Duffing's equation. Two formulations were examined: i) a generalized Galerkin method with Hermitian polynomials as interpolants both in spatial and temporal discretization (HHSE), ii) a mixed discontinuous Galerkin formulation with Hermitian cubic polynomials as interpolants for spatial discretization and Lagrangian spectral polynomials as interpolants for temporal discretization (HLSE). The first method revealed severe stability problems while the second method exhibited unconditional stability and was selected for detailed analysis. The spatialh-convergence rate of the HLSE method is of order α=p _s+1 (wherep _s is the spatial polynomial order). Temporalp-convergence of the HLSE method is exponential and theh-convergence rate based on the end points (the points corresponding to the final time of each element) is of order 2p _T−1 ≤α≤2p _T+1 (wherep _T is the temporal polynomial order). Due to the high accuracy of the HLSE method, good results were achieved for the cases considered using a relatively large spatial grid size (4 elements for first mode solutions) and a large integration time step (1/4 of the system period for first mode solutions, withp _T=3). All the first mode solution features were detected including the onset of the first period doubling bifurcation, the onset of chaos and the return to periodic motion. Two examples of second mode excitation produced homogeneous second mode and coupled first and second mode periodic solutions. Consequently, the STSE method is shown to be an accurate numerical method for simulation of nonlinear spatio-temporal dynamical systems exhibiting chaotic response.     

Mixed state-vector finite element analysis for a higher-order box beam theory

If thin-walled closed beams are analyzed by the standard Timoshenko beam elements, their structural behavior, especially near boundaries, cannot be accurately predicted because of the incapability of the Timoskenko theory to predict the sectional warping and distortional deformations. If a higher-order thin-walled box beam theory is used, on the other hand, accurate results comparable to those obtained by plate finite elements can be obtained. However, currently available two-node displacement based higher-order beam elements are not efficient in capturing exponential solution behavior near boundaries. Based on this motivation, we consider developing higher-order mixed finite elements. Instead of using the standard mixed formulation, we propose to develop the mixed formulation based on the state-vector form so that only the field variables that can be prescribed on the boundary are interpolated for finite element analysis. By this formulation, less field variables are used than by the standard mixed formulation, and the interpolated field variables have the physical meaning as the boundary work conjugates. To facilitate the discretization, two-node elements are considered. The effects of interpolation orders for the generalized stresses and displacements on the solution behavior are investigated along with numerical examples.     

Algorithmic formulation and numerical implementation of coupled electromagnetic‐inelastic continuum models for electromagnetic metal forming

The purpose of this work is the algorithmic formulation and implementation of a recent coupled electromagnetic‐inelastic continuum field model (Continuum Mech. Thermodyn. 2005; 17 :1–16) for a class of engineering materials, which can be dynamically formed using strong magnetic fields. Although in general relevant, temperature effects are for the applications of interest here minimal and are neglected for simplicity. In this case, the coupling is due, on the one hand, to the Lorentz force acting as an additional body force in the material. On the other hand, the spatio‐temporal development of the magnetic field is very sensitive to changes in the shape of the workpiece, resulting in additional coupling. The algorithmic formulation and numerical implementation of this coupled model is based on mixed‐element discretization of the deformation and electromagnetic fields combined with an implicit, staggered numerical solution scheme on two meshes. In particular, the mechanical degrees of freedom are solved on a Lagrangian mesh and the electromagnetic ones on an Eulerian one. The issues of the convergence behaviour of the staggered algorithm and the influence of data transfer between the meshes on the solution is discussed in detail. Finally, the numerical implementation of the model is applied to the modelling and simulation of electromagnetic sheet forming. Copyright © 2006 John Wiley & Sons, Ltd.     

A mortar formulation for 3D large deformation contact using NURBS-based isogeometric analysis and the augmented Lagrangian method

NURBS-based isogeometric analysis is applied to 3D frictionless large deformation contact problems. The contact constraints are treated with a mortar-based approach combined with a simplified integration method avoiding segmentation of the contact surfaces, and the discretization of the continuum is performed with arbitrary order NURBS and Lagrange polynomial elements. The contact constraints are satisfied exactly with the augmented Lagrangian formulation proposed by Alart and Curnier, whereby a Newton-like solution scheme is applied to solve the saddle point problem simultaneously for displacements and Lagrange multipliers. The numerical examples show that the proposed contact formulation in conjunction with the NURBS discretization delivers accurate and robust predictions. In both small and large deformation cases, the quality of the contact pressures is shown to improve significantly over that achieved with Lagrange discretizations. In large deformation and large sliding examples, the NURBS discretization provides an improved smoothness of the traction history curves. In both cases, increasing the order of the discretization is either beneficial or not influential when using isogeometric analysis, whereas it affects results negatively for Lagrange discretizations.     

On the computation of mode I and II thermal shock stress intensity factors using a boundary-only element method

A time domain boundary-only element method is used for the analysis of fractured planar bodies, subjected to thermal shock type loads. The uncoupled quasistatic thermoelasticity equations are solved without the need for domain discretization. The singular behaviour of the temperature and displacement fields, in the vicinity of the crack tip, is modelled by quarter-point elements. Transient thermal stress and heat flux intensity factors are evaluated from computed nodal values, using the well-known displacement and traction formulas. The accuracy of the method, the stability of the solution and the dependence on space-time discretization is investigated through comparison of present results with analytical and computational ones, taken from the literature. For mode I and mixed mode two-dimensional problems examined, the method proved to be a potent tool for fracture analysis in presence of severe thermal shock loads, even in the microcrack size range. Potent in the sense that, it is very stable and cost effective from the space-time discretization point of view.     

Flow through variably saturated soils

The equation of flow through variably saturated porous media is discretized via the Galerkin finite element formulation. The discretization is coupled with an approach for mesh generation and optimization of the node numbering scheme. Sensitivity analysis showed that the solution behavior is controlled by dimensionless quantities equivalent to Peclet and Courant numbers. For the form of equation investigated, no universal limiting values of Pe and Cr can be established because the values of these parameters depend on both the constitutive relations used and on initial conditions. For more efficient solution of the problem, a deformation scheme of the computational mesh is proposed, which accounts for the limiting Peclet and Courant numbers and for the shape of the deformed elements. Comparisons with other solutions showed that the numerical scheme performs very well.