Topology optimization of compliant mechanisms using element-free Galerkin method

This paper will propose a topology optimization approach for the design of large displacement compliant mechanisms with geometrical non-linearity by using the element-free Galerkin (EFG) method. In this method, the Shepard function is applied to construct a physically meaningful density approximant, to account for its non-negative and range-bounded property. Firstly, in terms of the original nodal density field, the Shepard function method functionally similar to a density filter is used to generate a non-local nodal density field with enriched smoothness over the design domain. The density of any node can be evaluated according to the nodal density variables located inside the influence domain of the interested node. Secondly, in the numerical implementation the Shepard function method is again employed to construct a point-wise density interpolant. Gauss quadrature is used to calculate the integration of background cells numerically, and the artificial densities over all Gauss points can be determined by the surrounding nodal densities within the influence domain of the concerned computational point. Finally, the moving least squares (MLS) method is applied to construct the shape functions using the weight functions with compact support for assembling the meshless approximations of state equations. Since MLS shape functions are lack of the Kronecker delta function property, the penalty method is applied to enforce the essential boundary conditions. A typical large-deformation compliant mechanism is used as the numerical example to demonstrate the effectiveness of the proposed method.     

A fast convergent rate preserving discontinuous Galerkin framework for rate-independent plasticity problems

In this paper a finite element framework based on the incomplete interior penalty Galerkin formulation, a non-symmetric discontinuous Galerkin method, is consistently formulated for modeling plasticity problems with small deformation. Because of its pure displacement-based framework, this proposed discontinuous Galerkin method is possibly able to completely preserve numerical integration algorithms efficiently developed in the traditional continuous Galerkin framework. Besides stresses on element interior quadrature points, stresses on element surface quadrature points are also required to return on yielding surfaces in this discontinuous Galerkin framework, which is able to provide more accurate material yielding profiles than the continuous Galerkin framework. The performance of the proposed discontinuous Galerkin framework has been evaluated in detail for J₂ and pressure-dependent plasticities using perfect plasticity, plasticity with hardening, and associative and non-associative material models. Quadratic convergent rates compatible to the tradition continuous Galerkin method for modeling plasticity problems have been achieved within a large penalty range in a nodal-based discontinuous Galerkin implementation.     

The element-free Galerkin method for the nonlinear p-Laplacian equation

The element-free Galerkin (EFG) method is developed in this paper for solving the nonlinear p-Laplacian equation. The moving least squares approximation is used to generate meshless shape functions, the penalty approach is adopted to enforce the Dirichlet boundary condition, the Galerkin weak form is employed to obtain the system of discrete equations, and two iterative procedures are developed to deal with the strong nonlinearity. Then, the computational formulas of the EFG method for the p-Laplacian equation are established. Numerical results are finally given to verify the convergence and high computational precision of the method.     

Materially and geometrically nonlinear analysis of laminated anisotropic plates by p-version of FEM

A p-version finite element model based on degenerate shell element is proposed for the analysis of orthotropic laminated plates. In the nonlinear formulation of the model, the total Lagrangian formulation is adopted with moderately large deflections and small rotations being accounted for in the sense of von Karman hypothesis. The material model is based on the Huber-Mises yield criterion and Prandtl-Reuss flow rule in accordance with the theory of strain hardening yield function, which is generalized for anisotropic materials by introducing the parameters of anisotropy. The model is also based on the equivalent-single layer laminate theory. The integrals of Legendre polynomials are used for shape functions with p-level varying from 1 to 10. Gauss-Lobatto numerical quadrature is used to calculate the stresses at the nodal points instead of Gauss points. The validity of the proposed p-version finite element model is demonstrated through several comparative points of view in terms of ultimate load, convergence characteristics, nonlinear effect, and shape of plastic zone.     

Finite element analysis of laminated shells of revolution

A finite element formulation for the analysis of axisymmetric fibre reinforced laminated shells subjected to axisymmetric load is presented. The formulation includes arbitrary number of bonded layers each of which may have different thicknesses, orientation of elastic axes, and elastic properties. Superparamatric curved elements[17] having four degrees of freedom per node including the normal rotation, are used. Stress-strain relation for an arbitrary layer is obtained from the consideration of three dimensional aspect of the problem. The element stiffness matrix has been obtained by using Gauss quadrature numerical integration, even though the elasticity matrix is different for different layers. The formulation is checked for a cylindrical tube subjected to internal pressure and axial tension, and the results are found to compare very well with the elastic solution [9].     

Element free method for static and free vibration analysis of spatial thin shell structures

The implementation of the element free Galerkin method (EFG) for spatial thin shell structures is presented in this paper. Both static deformation and free vibration analyses are considered. The formulation of the discrete system equations starts from the governing equations of stress resultant geometrically exact theory of shear flexible shells. Moving least squares approximation is used in both the construction of shape functions based on arbitrarily distributed nodes as well as in the surface approximation of general spatial shell geometry. Discrete system equations are obtained by incorporating these interpolations into the Galerkin weak form. The formulation is verified through numerical examples of static stress analysis and frequency analysis of spatial thin shell structures. For static load analysis, essential boundary conditions are enforced through penalty method and Lagrange multipliers while boundary conditions for frequency analysis are imposed through a weak form using orthogonal transformation techniques. The EFG results compare favorably with closed-form solutions and that of finite element analyses.     

2-D frictionless dynamic contact analysis of large deformable bodies by Petrov–Galerkin natural element method

This paper presents a Petrov–Galerkin natural element method for the nonlinear analysis of 2-D dynamic contact problems without friction. The frictionless dynamic contact problem is formulated according to the linearized total Lagrangian method incorporated with the linearized penalty method. The displacement increment is approximated with Laplace interpolation functions defined with the help of Voronoi polygons, while the admissible virtual displacement is expanded with constant strain functions which are supported on Delaunay triangles. The spatial numerical integration is carried out by applying the conventional Gauss quadrature rule to Delaunay triangles and the temporal time integration is performed by the implicit Newmark method. The validity of the proposed method is examined through the illustrating numerical experiments.     

Generalized Gaussian quadrature rules for discontinuities and crack singularities in the extended finite element method

New Gaussian integration schemes are presented for the efficient and accurate evaluation of weak form integrals in the extended finite element method. For discontinuous functions, we construct Gauss-like quadrature rules over arbitrarily-shaped elements in two dimensions without the need for partitioning the finite element. A point elimination algorithm is used in the construction of the quadratures, which ensures that the final quadratures have minimal number of Gauss points. For weakly singular integrands, we apply a polar transformation that eliminates the singularity so that the integration can be performed efficiently and accurately. Numerical examples in elastic fracture using the extended finite element method are presented to illustrate the performance of the new integration techniques.     

A nonconforming rotated Q1 approximation on tetrahedra

In this paper we construct an approximation that uses midpoints of edges on tetrahedra in three dimensions. The construction is based on the three-dimensional version of the rotated Q₁-approximation proposed by Rannacher and Turek (1992) [6]. We prove a priori error estimates for finite element solutions of the elasticity equations using the new element. Since it contains (rotated) bilinear terms it performs substantially better than the standard constant strain element in bending. It also allows for under-integration (in the form of one point Gauss integration of volumetric terms) in near incompressible situations. Numerical examples are included.     

Assessment and improvement of precise time step integration method

In this paper, the numerical stability and accuracy of Precise Time Step Integration Method are discussed in detail. It is shown that the method is conditionally stable and it has inherent algorithmic damping, algorithmic period error and algorithmic amplitude decay. However for discretized structural models, it is relatively easy for this time integration scheme to satisfy the stability conditions and required accuracy. Based on the above results, the optimum values of the truncation order L and bisection order N are presented. The Gauss quadrature method is used to improve the accuracy of the Precise Time Step Integration Method. Finally, two numerical examples are presented to show the feasibility of this improvement method.     

On the Quadrature and Weak Form Choices in Collocation Type Discontinuous Galerkin Spectral Element Methods

We examine four nodal versions of tensor product discontinuous Galerkin spectral element approximations to systems of conservation laws for quadrilateral or hexahedral meshes. They arise from the two choices of Gauss or Gauss-Lobatto quadrature and integrate by parts once (I) or twice (II) formulations of the discontinuous Galerkin method. We show that the two formulations are in fact algebraically equivalent with either Gauss or Gauss-Lobatto quadratures when global polynomial interpolations are used to approximate the solutions and fluxes within the elements. Numerical experiments confirm the equivalence of the approximations and indicate that using Gauss quadrature with integration by parts once is the most efficient of the four approximations.     

2-D large deformation analysis of nearly incompressible body by natural element method

Recently, the natural element method based on the Voronoi diagram and Delaunay triangulation has being intensively explored, as an advanced meshfree method. But, most of studies have been restricted to the linear elasticity problem. In this context, this study intends to explore its applicability to the non-linear solid mechanics, through the 2-D large deformation analysis of nearly incompressible bodies. The non-linear natural element approximation is implemented by the linearized total Lagrangian mixed u–p formulation incorporated with the penalty method. Numerical results illustrating the theoretical work are also presented.     

Penalty-finite element methods for the analysis of stokesian flows

A study of a class of finite element methods for the analysis of Stokes' problem based on the use of exterior penalty formulations is described. The effects of selective reduced integration (i.e., the use of quadrature rules for integrating the penalty terms which are of lower order than that required to integrate polynomial approximations of these terms exactly) are investigated. Error estimates are derived and the numerical stability of these methods, as depicted by a special Babuska-Brezzi condition, is explored in some detail. The results of several numerical experiments with these methods are also given.     

Solution of initial value problems by the differential quadrature method with Hermite bases

The differential quadrature method (DQM) is used to solve the first-order initial value problem. The initial condition is given at the beginning of the interval. The derivative of a space-independent variable at a sampling grid point within the interval can be defined as a weighted linear sum of the given initial conditions and the function values at the sampling grid points within the defined interval. Hermite polynomials have advantages compared with Lagrange and Chebyshev polynomials, and so, unlike other work, they are chosen as weight functions in the DQM. The proposed method is applied to a numerical example and it is shown that the accuracy of the quadrature solution obtained using the proposed sampling grid points is better than solutions obtained with the commonly used Chebyshev–Gauss–Lobatto sampling grid points.     

A high-order fast method for computing convolution integral with smooth kernel

In this paper we report on a high-order fast method to numerically calculate convolution integral with smooth non-periodic kernel. This method is based on the Newton-Cotes quadrature rule for the integral approximation and an FFT method for discrete summation. The method can have an arbitrarily high-order accuracy in principle depending on the number of points used in the integral approximation and a computational cost of O(Nlog(N)), where N is the number of grid points. For a three-point Simpson rule approximation, the method has an accuracy of O(h⁴), where h is the size of the computational grid. Applications of the Simpson rule based algorithm to the calculation of a one-dimensional continuous Gauss transform and to the calculation of a two-dimensional electric field from a charged beam are also presented.     

Analysis of 3D frame problems by DQEM using EDQ

The differential quadrature element method (DQEM) and extended differential quadrature (EDQ) have been proposed by the author. The development of a differential quadrature element analysis model of three-dimensional shear-undeformable frame problems adopting the EDQ is carried out. The element can be a nonprismatic beam. The EDQ technique is used to discretize the element-based governing differential equations, the transition conditions at joints and the boundary conditions on domain boundaries. An overall algebraic system can be obtained by assembling all of the discretized equations. A numerically rigorous solution can be obtained by solving the overall algebraic system. Mathematical formulations for the EDQ-based DQEM frame analysis are carried out. By using this DQEM model, accurate results of frame problems can efficiently be obtained. Numerical results demonstrate this DQEM model.     

Stability and accuracy of differential quadrature method in solving dynamic problems

In this paper, the differential quadrature method is used to solve dynamic problems governed by second-order ordinary differential equations in time. The Legendre, Radau, Chebyshev, Chebyshev–Gauss–Lobatto and uniformly spaced sampling grid points are considered. Besides, two approaches using the conventional and modified differential quadrature rules to impose the initial conditions are also investigated. The stability and accuracy properties are studied by evaluating the spectral radii and truncation errors of the resultant numerical amplification matrices. It is found that higher-order accurate solutions can be obtained at the end of a time step if the Gauss and Radau sampling grid points are used. However, the conventional approach to impose the initial conditions in general only gives conditionally stable time step integration algorithms. Unconditionally stable algorithms can be obtained if the modified differential quadrature rule is used. Unfortunately, the commonly used Chebyshev–Gauss–Lobatto sampling grid points would not generate unconditionally stable algorithms.     

A posteriori error estimates for a mixed-FEM formulation of a non-linear elliptic problem

We consider the numerical solution, via the mixed finite element method, of a non-linear elliptic partial differential equation in divergence form with Dirichlet boundary conditions. Besides the temperature u and the flux $\text{σ}$, we introduce ∇u as a further unknown, which yields a variational formulation with a twofold saddle point structure. We derive a reliable a posteriori error estimate that depends on the solution of a local linear boundary value problem, which does not need any equilibrium property for its solvability. In addition, for specific finite element subspaces of Raviart–Thomas type we are able to provide a fully explicit a posteriori error estimate that does not require the solution of the local problems. Our approach does not need the exact finite element solution, but any reasonable approximation of it, such as, for instance, the one obtained with a fully discrete Galerkin scheme. In particular, we suggest a scheme that uses quadrature formulas to evaluate all the linear and semi-linear forms involved. Finally, several numerical results illustrate the suitability of the explicit error estimator for the adaptive computation of the corresponding discrete solutions.     

A new kind of combinations between the Ritz-Galerkin and finite element methods for singularity problems

The coupling techniques of simplified hybrid plus penalty functions are first presented for matching the Ritz-Galerkin method and thek(k>-1)-order Lagrange finite element methods to solve complicated problems of elliptic equations, homogeneous or nonhomogeneous, in particular with singularities or unbounded domains. Optimal convergence rates of numerical solutions have been proved in the Sobolev norms. Moreover, the theoretical results obtained in this paper have been verified by numerical experiments for the singular Motz problem.     

Hermite and gauss type optimal quadratures for analytic functions

For integrals \(\int\limits_{ - 1}^\iota {w(x)f(x)dx} \) with a non-negative weight functionw(x) and analyticf, we develop Hermite and Gauss type optimal quadratures over the Hilbert spaceH ²(C_r) of functions analytic in a circle. Our development of these optimal quadratures in very similar to that of classical Hermite and Gauss quadratures; the role played by fundamental polynomials in the classical theory is replaced here by certain «fundamental rational functions». We then show, by arguments similar to those used in the classical case, that a Gauss type optimal quadrature has positive weights an abscissas lying in (?¹, 1).