A Trefftz approach to computational mechanics

This paper presents a Trefftz method for solving structural elasticity problems and flow problems of incompressible viscous fluids. The problem of unilateral contact is also dealt with. For each type of problem, Trefftz polynomials and associated variational formulations are given. Complex structures are studied by a sub‐structuring technique. This method requires the resolution of a non‐symmetrical linear system. It is shown that it is possible to take advantage of this Trefftz approximation in two ways: (i) the approach presented can be considered as a simplified method which enables a solution to be evaluated quickly; (ii) this approach also makes it possible to obtain a good quality solution associated with high degree polynomial bases. This method is adapted to optimization processes because the discretization of the structure requires only very few sub‐domains to build a good approximation and offers a great flexibility in use. Copyright © 2003 John Wiley & Sons, Ltd.     

A Kriging‐based error‐reproducing and interpolating kernel method for improved mesh‐free approximations

An error‐reproducing and interpolating kernel method (ERIKM), which is a novel and improved form of the error‐reproducing kernel method (ERKM) with the nodal interpolation property, is proposed. The ERKM is a non‐uniform rational B‐splines (NURBS)‐based mesh‐free approximation scheme recently proposed by Shaw and Roy (Comput. Mech. 2007; 40 (1):127–148). The ERKM is based on an initial approximation of the target function and its derivatives by NURBS basis functions. The errors in the NURBS approximation and its derivatives are then reproduced via a family of non‐NURBS basis functions. The non‐NURBS basis functions are constructed using a polynomial reproduction condition and added to the NURBS approximation obtained in the first step. In the ERKM, the interpolating property at the boundary is achieved by repeating the knot (open knot vector). However, for most problems of practical interest, employing NURBS with open knots is not possible because of the complex geometry of the domain, and consequently ERKM shape functions turn out to be non‐interpolating. In ERIKM, the error functions are obtained through localized Kriging based on a minimization of the squared variance of the estimate with the reproduction property as a constraint. Interpolating error functions so obtained are then added to the NURBS approximant. While enriching the ERKM with the interpolation property, the ERIKM naturally possesses all the desirable features of the ERKM, such as insensitivity to the support size and ability to reproduce sharp layers. The proposed ERIKM is finally applied to obtain strong and weak solutions for a class of linear and non‐linear boundary value problems of engineering interest. These illustrations help to bring out the relative numerical advantages and accuracy of the new method to some extent. Copyright © 2007 John Wiley & Sons, Ltd.     

A simple algorithm for localized construction of non‐matching structural interfaces

A simple and effective algorithm for the modular construction of non‐matched interfaces is presented for the partitioned solution of large‐scale structural problems. The formulation is based on a recently developed four‐field variational principle, which introduces a connection frame between the interfaced partitions. A key result of the present study is a frame nodal placement criterion that uniquely determines the frame discretization into piecewise linear elements so that the interface patch test condition is satisfied a priori. The method is demonstrated with several 2D and 3D example problems. Copyright © 2002 John Wiley & Sons, Ltd.     

A space–time discontinuous Galerkin method for the solution of the wave equation in the time domain

In recent years, the focus of research in the field of computational acoustics has shifted to the medium frequency regime and multiscale wave propagation. This has led to the development of new concepts including the discontinuous enrichment method. Its basic principle is the incorporation of features of the governing partial differential equation in the approximation. In this contribution, this concept is adapted for the simulation of transient problems governed by the wave equation. We present a space–time discontinuous Galerkin method with Lagrange multipliers, where the shape approximation in space and time is based on solutions of the homogeneous wave equation. The use of hierarchical wave‐like basis functions is enabled by means of a variational formulation that allows for discontinuities in both the spatial and the temporal discretizations. Numerical examples in one space dimension demonstrate the outstanding performance of the proposed method compared with conventional space–time finite element methods. Copyright © 2008 John Wiley & Sons, Ltd.     

The enhanced assumed strain method for the isogeometric analysis of nearly incompressible deformation of solids

Isogeometric analysis has recently become very popular for the numerical modeling of structures and fluids. Among other potential features, advantages of using a non‐uniform rational B‐splines (NURBS)‐based isogeometric analysis over the traditional finite element method include the possibility of using higher‐order polynomials for the basis functions of the approximation space, which may be easily built on a recursive (hierarchical) fashion as well as higher convergence ratio. Nevertheless, NURBS‐based isogeometric analysis suffers from the same problems depicted by other methods when it comes to reproduce isochoric deformations, that is, it shows volumetric locking, especially for low‐order basis functions. Similar remedies as those that have been proposed for the finite element method may be appropriate for integration in the NURBS‐based isogeometric analysis and some have already been tried with success. In this work, the analysis of the underlying space of incompressible deformations of a NURBS‐based isogeometric approximation is performed with the main objective of understanding the likelihood of volumetric locking. As a remedy, the enhanced assumed strain methodology is blended with the NURBS‐based isogeometric analysis to alleviate the volumetric locking associated with incompressible deformations. The solution includes a stabilization term derived directly from a penalized form of the classical Veubeke–Hu–Washizu three‐field variational principle. Copyright © 2012 John Wiley & Sons, Ltd.     

Non‐linear version of stabilized conforming nodal integration for Galerkin mesh‐free methods

A stabilized conforming (SC) nodal integration, which meets the integration constraint in the Galerkin mesh‐free approximation, is generalized for non‐linear problems. Using a Lagrangian discretization, the integration constraints for SC nodal integration are imposed in the undeformed configuration. This is accomplished by introducing a Lagrangian strain smoothing to the deformation gradient, and by performing a nodal integration in the undeformed configuration. The proposed method is independent to the path dependency of the materials. An assumed strain method is employed to formulate the discrete equilibrium equations, and the smoothed deformation gradient serves as the stabilization mechanism in the nodally integrated variational equation. Eigenvalue analysis demonstrated that the proposed strain smoothing provides a stabilization to the nodally integrated discrete equations. By employing Lagrangian shape functions, the computation of smoothed gradient matrix for deformation gradient is only necessary in the initial stage, and it can be stored and reused in the subsequent load steps. A significant gain in computational efficiency is achieved, as well as enhanced accuracy, in comparison with the mesh‐free solution using Gauss integration. The performance of the proposed method is shown to be quite robust in dealing with non‐uniform discretization. Copyright © 2002 John Wiley & Sons, Ltd.     

DSC‐Ritz method for the free vibration analysis of Mindlin plates

This paper introduces a novel method for the free vibration analysis of Mindlin plates. The proposed method takes the advantage of both the local bases of the discrete singular convolution (DSC) algorithm and the pb‐2 Ritz boundary functions to arrive at a new approach, called DSC‐Ritz method. Two basis functions are constructed by using DSC delta sequence kernels of the positive type. The energy functional of the Mindlin plate is represented by the newly constructed basis functions and is minimized under the Ritz variational principle. Extensive numerical experiments are considered by different combinations of boundary conditions of Mindlin plates of rectangular and triangular shapes. The performance of the proposed method is carefully validated by convergence analysis. The frequency parameters agree very well with those in the literature. Numerical experiments indicate that the proposed DSC‐Ritz method is a very promising new method for vibration analysis of Mindlin plates. Copyright © 2004 John Wiley & Sons, Ltd.     

B-spline approximation in boundary face method for three-dimensional linear elasticity

In this paper, basis functions generated from B-spline or Non-Uniform Rational B-spline (NURBS), are used for approximating the boundary variables to solve the 3D linear elasticity Boundary Integral Equations (BIEs). The implementation is based on the BFM framework in which both boundary integration and variable approximation are performed in the parametric spaces of the boundary surfaces to keep the exact geometric information in the BIEs. In order to reduce the influence of tensor product of B-spline and make the discretization of a body surface easier, the basis functions defined in global intervals are translated into local form. B-spline fitting function built with the local basis functions is converted into an interpolation type of function in which the nodal values of the boundary variables are used for control points. Numerical tests for 3D linear elasticity problems show that the BFM with B-spline basis functions outperforms that with the well-known Moving Least Square (MLS) approximation.     

Implementation of a symmetric boundary element method in transient heat conduction with semi‐analytical integrations

A time‐convolutive variational hypersingular integral formulation of transient heat conduction over a 2‐D homogeneous domain is considered. The adopted discretization leads to a linear equation system, whose coefficient matrix is symmetric, and is generated by double integrations in space and time. Assuming polynomial shape functions for the boundary unknowns, a set of compact formulae for the analytical time integrations is established. The spatial integrations are performed numerically using very efficient formulae just recently proposed. The competitiveness from the computational point of view of the symmetric boundary integral equation approach proposed herein is investigated on the basis of an original computer implementation. Copyright © 1999 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.     

A quasi‐interpolation method for solving stiff ordinary differential equations

Based on the idea of quasi‐interpolation and radial basis functions approximation, a numerical method is developed to quasi‐interpolate the forcing term of differential equations by using radial basis functions. A highly accurate approximation for the solution can then be obtained by solving the corresponding fundamental equation and a small size system of equations related to the initial or boundary conditions. This overcomes the ill‐conditioning problem resulting from using the radial basis functions as a global interpolant. Error estimation is given for a particular second‐order stiff differential equation with boundary layer. The result of computations indicates that the method can be applied to solve very stiff problems. With the use of multiquadric, a special class of radial basis functions, it has been shown that a reasonable choice for the optimal shape parameter is obtained by taking the same value of the shape parameter as the perturbed parameter contained in the stiff equation. Copyright © 2000 John Wiley & Sons, Ltd.     

Static deformations and vibration analysis of composite and sandwich plates using a layerwise theory and multiquadrics discretizations

In this paper, the static and free vibration analysis of composite plates are performed, using a layerwise deformation theory and multiquadrics discretization. This meshless discretization method considers radial basis functions as the approximation method for both the equations of motion and the boundary conditions. The combination of this layerwise theory and the multiquadrics discretization method allows a very accurate prediction of the natural frequencies.     

Numerical stabilization of Biot's consolidation model by a perturbation on the flow equation

In this paper a stabilized finite element scheme for the poroelasticity equations is proposed. This method, based on the perturbation of the flow equation, allows us to use continuous piecewise linear approximation spaces for both displacements and pressure, obtaining solutions without oscillations independently of the chosen discretization parameters. The perturbation term depends on a parameter which is established in terms of the mesh size and the properties of the material. In the one‐dimensional case, this parameter is shown to be optimal. Some numerical experiments are presented indicating the efficiency of the proposed stabilization technique. Copyright © 2008 John Wiley & Sons, Ltd.     

A new wavelet-based thin plate element using B-spline wavelet on the interval

By interacting and synchronizing wavelet theory in mathematics and variational principle in finite element method, a class of wavelet-based plate element is constructed. In the construction of wavelet-based plate element, the element displacement field represented by the coefficients of wavelet expansions in wavelet space is transformed into the physical degree of freedoms in finite element space via the corresponding two-dimensional C₁ type transformation matrix. Then, based on the associated generalized function of potential energy of thin plate bending and vibration problems, the scaling functions of B-spline wavelet on the interval (BSWI) at different scale are employed directly to form the multi-scale finite element approximation basis so as to construct BSWI plate element via variational principle. BSWI plate element combines the accuracy of B-spline functions approximation and various wavelet-based elements for structural analysis. Some static and dynamic numerical examples are studied to demonstrate the performances of the present element.     

A hybrid variational‐collocation immersed method for fluid‐structure interaction using unstructured T‐splines

We present a hybrid variational‐collocation, immersed, and fully‐implicit formulation for fluid‐structure interaction (FSI) using unstructured T‐splines. In our immersed methodology, we define an Eulerian mesh on the whole computational domain and a Lagrangian mesh on the solid domain, which moves arbitrarily on top of the Eulerian mesh. Mathematically, the problem reduces to solving three equations, namely, the linear momentum balance, mass conservation, and a condition of kinematic compatibility between the Lagrangian displacement and the Eulerian velocity. We use a weighted residual approach for the linear momentum and mass conservation equations, but we discretize directly the strong form of the kinematic relation, deriving a hybrid variational‐collocation method. We use T‐splines for both the spatial discretization and the information transfer between the Eulerian mesh and the Lagrangian mesh. T‐splines offer us two main advantages against non‐uniform rational B‐splines: they can be locally refined and they are unstructured. The generalized‐α method is used for the time discretization. We validate our formulation with a common FSI benchmark problem achieving excellent agreement with the theoretical solution. An example involving a partially immersed solid is also solved. The numerical examples show how the use of T‐junctions and extraordinary nodes results in an accurate, efficient, and flexible method. Copyright © 2015 John Wiley & Sons, Ltd.     

Overview and construction of meshfree basis functions: from moving least squares to entropy approximants

In this paper, an overview of the construction of meshfree basis functions is presented, with particular emphasis on moving least‐squares approximants, natural neighbour‐based polygonal interpolants, and entropy approximants. The use of information‐theoretic variational principles to derive approximation schemes is a recent development. In this setting, data approximation is viewed as an inductive inference problem, with the basis functions being synonymous with a discrete probability distribution and the polynomial reproducing conditions acting as the linear constraints. The maximization (minimization) of the Shannon–Jaynes entropy functional (relative entropy functional) is used to unify the construction of globally and locally supported convex approximation schemes. A JAVA applet is used to visualize the meshfree basis functions, and comparisons and links between different meshfree approximation schemes are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

A domain decomposition method for discontinuous Galerkin discretizations of Helmholtz problems with plane waves and Lagrange multipliers

A nonoverlapping domain decomposition (DD) method is proposed for the iterative solution of systems of equations arising from the discretization of Helmholtz problems by the discontinuous enrichment method. This discretization method is a discontinuous Galerkin finite element method with plane wave basis functions for approximating locally the solution and dual Lagrange multipliers for weakly enforcing its continuity over the element interfaces. The primal subdomain degrees of freedom are eliminated by local static condensations to obtain an algebraic system of equations formulated in terms of the interface Lagrange multipliers only. As in the FETI‐H and FETI‐DPH DD methods for continuous Galerkin discretizations, this system of Lagrange multipliers is iteratively solved by a Krylov method equipped with both a local preconditioner based on subdomain data, and a global one using a coarse space. Numerical experiments performed for two‐ and three‐dimensional acoustic scattering problems suggest that the proposed DD‐based iterative solver is scalable with respect to both the size of the global problem and the number of subdomains. Copyright © 2009 John Wiley & Sons, Ltd.     

Variational boundary-integral-equation approach to unilateral contact problems in elasticity

On the basis of the boundary integral equation method, three variational principles for the frictionless unilateral contact problem in elasticity are presented. Two of them are saddle-point principles for the boundary unknowns (including the contact displacements); a third one is a maximum principle for the unknown contact displacements only. A discretization by boundary elements leads to algebraic formulations in the shape either of quadratic programming problems, or of linear complementarity problems, all characterized by symmetry and sign definiteness of the coefficient matrices. The method is also applicable to contact problems between two uncompenetrable elastic solids, as well as to the crack problem of fracture mechanics.     

基于最小转换能原理的一种逐步积分法

基于Benthien_Gurtin 给出的拉普拉斯空间上相应的最小转换能原理,将拉普拉斯空间上的泛函恢复到原空间后给出一个只含单重卷积形式的泛函。对此泛函相继进行空间和时间域离散分别给出了新的泛函。采用含非时间步参数的三次Hermite 插值函数逼近节点位移,给出了一种用于动力响应分析的无条件稳定逐步积分算法。通过对算法的稳定性研究,选取了参数的合适值。精度分析和数值算例表明提出的方法是一种可以应用于工程实际并具有较高精度的方法。     

A method for solving partial differential equations via radial basis functions: Application to the heat equation

In this work a technique is proposed for solving partial differential equations using radial basis functions. The approach is different from the traditional schemes. The radial basis functions are very suitable instruments for solving partial differential equations of various types. However, the matrices which result from the discretization of the equations are usually ill-conditioned especially in higher-dimensional problems. In the current paper, a stable method will be proposed for solving the partial differential equations and will be generalized to solve higher-dimensional problems. To the contrast of most existing methods, the new technique provides a closed form approximation for the solution. Another advantage of the developed method is that it can be applied to problems with nonregular geometrical domains.