Second-order shape design sensitivity using P-version finite element analysis

Thep-version finite element analysis (FEA) approach is attractive for design sensitivity analysis (DSA) and optimization due to its high accuracy of analysis results, even with coarse mesh; insensitivity to finite element mesh distortion and aspect ratio; and tolerance for large shape design changes during design iterations. A continuum second-order shape DSA formulation is derived and implemented usingp-version FEA. The second-order shape design sensitivity can be used for reliability based analysis and design optimization by incorporating it with the second-order reliability analysis method (SORM). Both the second-order shape DSA formulations with respect to the single and mixed shape design parameters are derived for elastic solids using the material derivative concept. Both the direct differentiation and hybrid methods are presented in this paper. A shape DSA is implemented by using an establishedp-version FEA code, STRESS CHECK. Two numerical examples, a connecting rod and bracket, are presented to demonstrate the feasibility and accuracy of the proposed seond-order shape DSA approach.     

A multiscale formulation of the Discontinuous Petrov–Galerkin method for advective–diffusive problems

We consider the Discontinuous Petrov–Galerkin method for the advection–diffusion model problem, and we investigate the application of the variational multiscale method to this formulation. We show the exact modeling of the fine scale modes at the element level for the linear case, and we discuss the approximate modeling both in the linear and in the non-linear cases. Furthermore, we highlight the existing link between this multiscale formulation and the p-version of the finite element method. Numerical examples illustrate the behavior of the proposed scheme.     

Shape sensitivity analysis using low- and high-order finite elements

A general procedure to perform the sensitivity analysis for the shape optimal design of elastic structures is proposed. The method is based on the implicit differentiation of the discretized equilibrium equations used in the finite element method (FEM). The so-called semianalytical approach is followed, that is, finite differences are used to differentiate the finite element matrices. The technique takes advantage of the geometric modeling concepts typical of the computer-aided design (CAD) technology used in the creation of a compact design model. This procedure is largely independent of the types of finite elements used in the analysis and has been implemented in ah-version andp-version finite element program. Very accurate and stable shape sensitivity derivatives were obtained from both programs over a wide range of finite difference step sizes. It is shown that the method is computationally efficient, general, and relatively easy to implement. Some classical shape optimal design problems have been solved using the CONLIN optimizer supplied with these gradients.     

High order finite elements for shells

In this article the p-version finite element method is applied to thin-walled structures. Two different hierarchic element formulations are compared, a shell approach as well as a shell-like, solid formulation. Both approaches are compared for linear elastic and elastoplastic problems. Special emphasis is placed on the efficiency as well as on determining the area of application for both formulations.     

Mixed least-squares finite element model for the static analysis of laminated composite plates

This paper presents a mixed finite element model for the static analysis of laminated composite plates. The formulation is based on the least-squares variational principle, which is an alternative approach to the mixed weak form finite element models. The mixed least-squares finite element model considers the first-order shear deformation theory with generalized displacements and stress resultants as independent variables. Specifically, the mixed model is developed using equal-order C⁰ Lagrange interpolation functions of high p-levels along with full integration. This mixed least-squares-based discrete model yields a symmetric and positive-definite system of algebraic equations. The predictive capability of the proposed model is demonstrated by numerical examples of the static analysis of four laminated composite plates, with different boundary conditions and various side-to-thickness ratios. Particularly, the mixed least-squares model with high-order interpolation functions is shown to be insensitive to shear-locking.     

A meshless FPM model for solving nonlinear material problems with proportional loading based on deformation theory

In this work a methodology of meshless finite points method for the analysis of nonlinear material problems with proportional loading based on deformation theory is presented.In finite points method the approximation around each point is obtained by using weighted least square techniques. The discrete system of equation is constructed by means of a point collocation procedure. The non-dependence on a mesh or integration procedures is an important aspect which transforms the finite point method in a truly meshless technique.Hencky’s total deformation theory and an elastic approach is used on the determination of stress–strain fields. This approach introduces the concept of effective material properties which are considered as spatial field variables and to be functions of equilibrium stress state and material properties.The present results are in good agreement with those obtained by nonlinear finite element method and previous work in this meshless context. Nevertheless the present methodology is based on a strong formulation, keeping the meshless characteristics of FPM.     

Design curve determination for two-layered wire rope strand using p-version finite element code

Parametric analysis of a two-layered axially loaded strand is performed using the recently developed p-version finite element code, which describes the geometry well and takes into account all possible inter-wire motions and frictional contact between the wires. A special nonlinear contact theory was developed based on the Hertz-theory. It is assumed that the wires have homogenous, isotropic, linear elastic material properties. The developed code is a tool for designing wire rope strands that require low computer resources and short computational time. Case studies are performed to verify and demonstrate the efficiency and applicability of the method. Design curves are presented according to the strand geometry parameters such as helix angle and ratio of the wire radius in the different layers. The optimal geometry parameters for a given strand can be determined using these design curves.     

Implementation of a C1 triangular element based on the P-version of the finite element method

The implementation of a computer code CONE (for C¹ continuity) based on the p-version of the finite element method is described. A hierarchic family of triangular finite elements of degree p ≥ 5 is used. This family enforces C¹-continuity across inter-element boundaries, and the code is applicable to fourth order partial differential equations in two independent variables, in particular to the biharmonic equation. Applications to several benchmark problems in plate bending are presented. Sample results are examined and compared both with theoretical predictions and with the computations of other programs. Significant improvements are shown for the results obtained using CONE.     

<Emphasis Type="Italic">hp</Emphasis>-Version <Emphasis Type="Italic">a priori</Emphasis> Error Analysis of Interior Penalty Discontinuous Galerkin Finite Element Approximations to the Biharmonic Equation

We consider the symmetric formulation of the interior penalty discontinuous Galerkin finite element method for the numerical solution of the biharmonic equation with Dirichlet boundary conditions in a bounded polyhedral domain in . For a shape-regular family of meshes consisting of parallelepipeds, we derive hp-version a priori bounds on the global error measured in the L² norm and in broken Sobolev norms. Using these, we obtain hp-version bounds on the error in linear functionals of the solution. The bounds are optimal with respect to the mesh size h and suboptimal with respect to the degree of the piecewise polynomial approximation p. The theoretical results are confirmed by numerical experiments, and some practical applications in Poisson–Kirchhoff thin plate theory are presented.