Accuracy Analysis of the Semi-Analytical Method for Shape Sensitivity Calculation∗

ABSTRACT

The semi-analytical method of design sensitivity analysis that is widely used for calculating derivatives of static response with respect to design variables for structures modeled by finite elements is studied in this paper. It is shown that the method can have serious accuracy problems for shape design variables in structures modeled by beam, plate, truss, frame, and solid elements. Errors are shown to be associated with an incompatibility of the sensitivity field with the structure. An error index is developed to test the accuracy of the semi-analytical method. It characterizes the difference in errors between a general finite difference method and the semi-analytical method. A method for improving the accuracy of the semi-analytical method (when possible) is provided. Examples are presented to demonstrate the use of the error index.     

基于改进位移模式的一维有限元超收敛算法

将多尺度方法的思想与超收敛计算的解析公式结合起来,提出了改进有限元位移模式的算法。利用超收敛计算的解析公式,将高阶有限元解的位移模式用常规有限元解的位移模式表示。用常规有限元解的位移模式与高阶有限元解的位移模式之和构造新的位移模式,采用积分形式推导了单元刚度矩阵。该算法在前处理和后处理两个阶段都使用超收敛计算公式,在常规试函数的基础上,增加了高阶试函数,使得单元内平衡方程的残差减少,从而达到提高精度的目标。对于线性单元,本文结点和单元的位移、导数都达到了h4阶的超收敛精度。     

Implicit Calculations of an Aeroelasticity Problem

The purpose of this work is to show that a linearized implicit scheme for the flow resolution can be an efficient and accurate method for solving fluid-structure interaction. The fluid is modeled by the Euler equations in two dimensions and the structure by a one (free piston) or a two (NACA0012 airfoil) degrees of freedom system. The schemes are developed using a finite volume/finite element formulation and, stating the moving boundary problem in the space-time domain, the Riemann solver is generalized in a suitable manner. Assuming a modal decomposition for the structure's response, an analytical solution to the equation of motion is obtained.

The effects of the linearized implicit scheme on the aeroelastic response are demonstrated on the free piston and the NACA 0012 airfoil problems. In the latter case, we focus on the capability of the linearized implicit scheme to accurately predict the stability limit of the coupled response (wing flutter analysis). Although the above analysis is performed using a rigid transformation, a robust moving mesh strategy is presented for more general 2-D and 3-D deformations.     

Hermitian space-time finite elements for estuarine mass transport

Space-time finite element solutions of the convection–dispersion equation using higher-order nodal continuity and Hermitian polynomial shape functions are described. Five separate elements ranging from a complete linear element with C^0,0 nodal continuity to a complete first-order Hermitian element with C^1,1 nodal continuity are subjected to detailed analysis. Wave deformation analyses identify the source of leading or trailing edge oscillations, trailing edge oscillations being the major source of difficulty. These observations are confirmed by numerical experiments which further demonstrate the potential of higher-order nodal continuity. The performance of the complete first-order Hermitian element is quite satisfactory and measurably superior to the linear element.     

Three-dimensional streamlined finite elements: Design of extrusion dies

A method to determine three-dimensional die shapes from extrudate swell and vice versa is presented using a three-dimensional Galerkin finite element method based on a streamlined formulation with the fluid velocities and pressures represented by triquadratic and trilinear basis functions respectively. The three-dimensional streamlined method, an extension of the two-dimensional formulation, uses successive streamsurfaces to form a boundary-conforming co-ordinate system. This produces a fixd, computational domain leaving the spatial location of the elements as unknowns to be determined with the standard primary variables (u, v, w, p). The extrudate produced by a die of a given shape is considered for moderate Reynolds numbers. Finally, the method is extended to address the problem of die design, where a die profile is sought to produce a target extrudate shape.     

基于改进位移模式的一维C1有限元超收敛算法

提出了基于改进位移模式的一维C1有限元超收敛算法。利用单元内部需满足平衡方程的条件,推导了超收敛计算解析公式的显式,即将高阶有限元解的位移模式用常规有限元解的位移模式表示。用常规有限元解的位移模式与高阶有限元解的位移模式之和构造新的位移模式。采用积分形式推导了单元刚度矩阵。该算法在前处理阶段使用了超收敛计算公式,在常规试函数的基础上,增加了高阶试函数,使得单元内平衡方程的残差减少,从而达到提高精度的目标。对于Hermite单元,本文的结点和单元的位移、导数都达到了h4阶的超收敛精度。     

Synthesis of highly convergent 2D and 3D finite elements for short acoustic wave simulation

New higher-order finite elements of enhanced convergence properties for acoustic wave simulation are presented in the paper. The element matrices are obtained by combining modal synthesis and optimization techniques in order to achieve minimum errors of higher modes of the computational domain. As a result, simulation models of propagating wave pulses require a smaller number of finite element divisions per wavelength compared to the conventional element model thus significantly reducing computational costs. Though finite element matrices are obtained in optimization, the resulting patterns of the matrices are versatile and further can be used in any wave propagation model. The mass matrices of the elements are diagonal, so explicit time integration schemes are applicable. The usage of new elements is especially efficient in situations where wavelengths of the simulated signal are much shorter than the dimensions of the computational domain. This is referred to as short wave propagation analysis. The results of wave propagation simulation for ultrasonic measurements are presented as application examples. The B-scans and computed dispersion curves are provided for visual interpretation of the results.     

Accurate transverse stress evaluation in composite/sandwich thick laminates using a C0 HSDT and a novel post-processing technique

An efficient method for accurate evaluation of through-the-thickness distribution of transverse stresses in thick composite and sandwich laminates, using a displacement-based C⁰ higher-order shear deformation theory (HSDT), is presented. The technique involves a least square of error (LSE) method applied to the 3D equilibrium equations at the post-processing phase, after a primary finite element analysis is performed using the HSDT. This is distinctly different from the conventional method of integrating the 3D equilibrium equations, for transverse stress recovery in composite laminates during post-processing. Competence of the technique is demonstrated in the numerical examples through comparison with results from first-order shear deformation theory (FSDT), another HSDT and those from analytical and 3D elasticity solutions available in literature.     

THE APPLICATION OF A FINITE DIFFERENCE/FINITE VOLUME ALGORITHM TO SOLVE MAXWELL'S EQUATIONS IN THE TIME DOMAIN

Abstract

A finite volume/finite difference method based on Ni's multigrid formulation is introduced for the solution of Maxwell's equations. The scheme is presented for the cases of transverse magnetic scattering from two-dimensional circular and square cylinders, as well as from NACA 0012 airfoil. The codes are validated against the traditional Method of Moments, which is analogous to a panel method in CFD. The circular cylinder scattering is compared to the analytical series solution for better understanding how the roles of numerical dispersion and dissipation errors affect the solution. The reflecting boundary conditions are modeled by the idea of inducing fields inside the conductor and a method of modeling the singularities that arise at a sharp corner is presented. Absorbing boundary conditions are modeled by integrating along the characteristic compatibility equations in the direction of the outgoing wave.     

A Yield Surface Linearization Procedure in Limit Analysis∗

ABSTRACT

Within the framework of discretized structural models and piecewise linear yield polyhedra, a heuristic is presented to iteratively improve the limit load estimation by approaching locally the plastic admissibility domain. For the two-dimensional active stress situation, the scheme requires the generation of at the most three vertices, in addition to the initial four, to describe the improved inscribed yield polygon. Two examples are given to illustrate the procedure and to clarify various aspects of the algorithm presented.     

Modeling quasi-static crack growth with the extended finite element method Part I: Computer implementation

The extended finite element method (X-FEM) is a numerical method for modeling strong (displacement) as well as weak (strain) discontinuities within a standard finite element framework. In the X-FEM, special functions are added to the finite element approximation using the framework of partition of unity. For crack modeling in isotropic linear elasticity, a discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are used to account for the crack. This enables the domain to be modeled by finite elements without explicitly meshing the crack surfaces, and hence quasi-static crack propagation simulations can be carried out without remeshing. In this paper, we discuss some of the key issues in the X-FEM and describe its implementation within a general-purpose finite element code. The finite element program Dynaflow™ is considered in this study and the implementation for modeling 2-d cracks in isotropic and bimaterial media is described. In particular, the array-allocation for enriched degrees of freedom, use of geometric-based queries for carrying out nodal enrichment and mesh partitioning, and the assembly procedure for the discrete equations are presented. We place particular emphasis on the design of a computer code to enable the modeling of discontinuous phenomena within a finite element framework.     

Design of a tuned vibration absorber for a slender hollow cylindrical structure

Abstract

In this paper, details of the design work for a tuned vibration absorber to be used on a hollow cylindrical structure is presented. The vibration problem is of resonant type and the tuned vibration absorber is designed to suppress the displacement vibration response of the free end of the slender hollow structure dominated by the contribution of its lowest transverse vibration modes. The structure is modeled using a commercial finite element software. Finite element model of the structure is verified using experimentally obtained frequency response functions and modal parameters. Effective parameters of the tuned vibration absorber design are then determined based on finite element analysis simulations of the vibration suppression performance of the tuned vibration absorber as it is used on the structure. Details of the tuned vibration absorber design are determined and a prototype is fabricated. Prototype tuned vibration absorber is then characterized experimentally both as a standalone system and also as it is used on the main structure. Vibration reduction performance of the physical prototype of the tuned vibration absorber is also compared with its vibration reduction performance estimated from finite element analysis simulations so that the analysis based design process can be validated.

Communicated by Dumitru Caruntu.     

A finite deformation theory of higher-order gradient crystal plasticity

For higher-order gradient crystal plasticity, a finite deformation formulation is presented. The theory does not deviate much from the conventional crystal plasticity theory. Only a back stress effect and additional differential equations for evolution of the geometrically necessary dislocation (GND) densities supplement the conventional theory within a non-work-conjugate framework in which there is no need to introduce higher-order microscopic stresses that would be work-conjugate to slip rate gradients. We discuss its connection to a work-conjugate type of finite deformation gradient crystal plasticity that is based on an assumption of the existence of higher-order stresses. Furthermore, a boundary-value problem for simple shear of a constrained thin strip is studied numerically, and some characteristic features of finite deformation are demonstrated through a comparison to a solution for the small deformation theory. As in a previous formulation for small deformation, the present formulation applies to the context of multiple and three-dimensional slip deformations.     

Upwind basis finite elements for convection-dominated problems

Finite elements using higher-order basis functions in the spirit of the QUICK method for convection-dominated fluid flow and transport problems are introduced and demonstrated. Instead of introducing new internal degrees of freedom, completeness is achieved by including functions based on nodal values exterior and upwind to the element domain. Applied with linear test functions to the weak statements for convection-dominated problems, a family of Petrov–Galerkin finite elements is developed. Quadratic and cubic versions are demonstrated for the one-dimensional convection–diffusion test problem. Elements of up to seventh degree are used for local solution refinement. The behaviour of these elements for one-dimensional linear and non-linear advection is investigated. A two-dimensional quadratic upwind element is demonstrated in a streamfunction–vorticity formulation of the Navier–Stokes equations for a driven cavity flow test problem. With some minor reservations, these elements are recommended for further study and application.     

A Direct Method for the Generation of Shape Functions of Any Desired Continuity for Rectangular Finite Elements

Abstract

A direct method is presented for the generation of shape functions for rectangular finite elements, with any desired continuity of the shape functions across the interelement boundaries, i.e., the shape functions can be of any prescribed class C^N, N = 0, 1, 2, 3 …. The method is illustrated with seven examples, four of which are previously published elements. The other three examples represent new elements, two of which are conforming C¹ elements, and the third one is of class C². One of the C1 elements and the C² element reproduce exactly all the terms up to the sextic polynomial.     

Delamination of laminated composite plates by p-convergent partial discrete-layer elements with VCCT

The simulation of the delamination process in laminated composite plates is quite complex and requires advanced finite element modeling techniques. Failure analysis tools must be able to predict initiation, size and propagation of delamination process. This paper presents the p-convergent partial discrete-layer elements with the virtual crack closure technique (VCCT) for the delamination analysis of laminated composite plates. The proposed element can be formulated by the suitable dimensional reduction from three-dimensional solid to two-dimensional plate. It is assumed that the piecewise linear variation of in-plane displacements and the constant value of out-of-plane displacements across the thickness. The higher-order approximation based on integrals of Legendre polynomials is used to define displacement fields. The three-dimensional VCCT is also slightly modified to incorporate with the proposed elements to estimate the energy release rate. The initiation of delamination occurs when the energy release rate for a displacement increment is same as the critical energy release rate corresponding to fracture toughness. The approach is to use a fracture mechanics criterion, but to avoid the complex moving mesh technique. At first, the validation and characteristic of the proposed elements are investigated on isotropic plates and orthotropic laminated plates, compared with referenced values. Then for fracture analysis, the efficiency of proposed approach is demonstrated with the help of additionally two problems such as the double-cantilever-beam test and the orthotropic laminated square plate with interior delamination.     

C~0型高阶锯齿理论及复合材料厚梁热应力分析

基于已有锯齿理论构造单元时,需使用满足单元间C1连续的插值函数,难于构造多节点高阶单元,而且精度较低。针对已有锯齿理论存在的问题,本文首先发展了C0型锯齿理论。通过虚位移原理推导出在热载荷作用下复合材料梁的平衡方程,并给出了简支复合材料层合梁解析解。基于发展的锯齿理论分析了复合材料夹层梁和层合梁热膨胀问题,并与其他理论结果对比。数值结果表明,发展的C0型锯齿理论能克服已有锯齿理论的难题。     

采用非协调元的连续体拓扑优化设计

介绍了满足分片检验条件的一种非协调元，推导了结构拓扑优化设计中数值计算和敏度分析的基本方程，给出了数值算例，并对协调等参元和非协调元的拓扑优化结果进行了对照，最后的优化结果表明：采用非协调元所得的优化解已经能够使用，如果再实施过滤技术，设计区域中的中间密度单元明显减少，优化结果会更加精致；使用两类单元的求解效率和优化迭代次数相近；非协调元比等参元具有更高精度的拓扑优化结果。能进一步克服棋盘格式。     

Construction of n-sided polygonal spline element using area coordinates and B-net method

In general, triangular and quadrilateral elements are commonly applied in two-dimensional finite element methods. If they are used to compute polycrystalline materials, the cost of computation can be quite significant. Polygonal elements can do well in simulation of the materials behavior and provide greater flexibility for the meshing of complex geometries. Hence, the study on the polygonal element is a very useful and necessary part in the finite element method. In this paper, an n-sided polygonal element based on quadratic spline interpolant, denoted by PS2 element, is presented using the triangular area coordinates and the B-net method. The PS2 element is conforming and can exactly model the quadratic field. It is valid for both convex and non-convex polygonal element, and insensitive to mesh distortions. In addition, no mapping or coordinate transformation is required and thus no Jacobian matrix and its inverse are evaluated. Some appropriate examples are employed to evaluate the performance of the proposed element.