Showing papers on "Direct stiffness method published in 2006"

A systematic analytical method for PKM stiffness matrix calculation

Abstract: The purpose of this work is to propose a new approach for the calculation of PKM stiffness matrix by using an analytical method based on matrix structural analysis. This method has as the main advantage to be systematic and it also can be applied to hyper-static PKM stiffness analysis. The implementation of the proposed method is fast and convenient and it can easily be involved during PKM design optimization. Moreover, as the stiffness matrix is obtained in a close form, it can be implemented directly on the robot controller. In other words, the proposed method could be used to reduce the gap existing between actual PKM and those that have to meet the high accuracy specifications required for machining applications. At first this paper presents the proposed analytical method that is applied to calculate the stiffness matrix of a delta parallel structure. Then the experimental results that have been done to evaluate and validate its efficiency are presented

Topology optimization of 2D continua for minimum compliance using parallel computing

Abstract: Topology optimization is often used in the conceptual design stage as a preprocessing tool to obtain overall material distribution in the solution domain. The resulting topology is then used as an initial guess for shape optimization. It is always desirable to use fine computational grids to obtain high-resolution layouts that minimize the need for shape optimization and postprocessing (Bendsoe and Sigmund, Topology optimization theory, methods and applications. Springer, Berlin Heidelberg New York 2003), but this approach results in high computation cost and is prohibitive for large structures. In the present work, parallel computing in combination with domain decomposition is proposed to reduce the computation time of such problems. The power law approach is used as the material distribution method, and an optimality criteria-based optimizer is used for locating the optimum solution [Sigmund (2001)21:120–127; Rozvany and Olhoff, Topology optimization of structures and composites continua. Kluwer, Norwell 2000]. The equilibrium equations are solved using a preconditioned conjugate gradient algorithm. These calculations have been done using a master–slave programming paradigm on a coarse-grain, multiple instruction multiple data, shared-memory architecture. In this study, by avoiding the assembly of the global stiffness matrix, the memory requirement and computation time has been reduced. The results of the current study show that the parallel computing technique is a valuable tool for solving computationally intensive topology optimization problems.

Hybrid compliance-stiffness matrix method for stable analysis of elastic wave propagation in multilayered anisotropic media.

Abstract: This paper presents the hybrid compliance-stiffness matrix method for stable analysis of elastic wave propagation in multilayered anisotropic media. The method utilizes the hybrid matrix of each layer in a recursive algorithm to deduce the stack hybrid matrix for a multilayered structure. Like the stiffness matrix method, the hybrid matrix method is able to eliminate the numerical instability of transfer matrix method. By operating with total stresses and displacements, it also preserves the convenience for incorporating imperfect or perfect interfaces. However, unlike the stiffness matrix, the hybrid matrix remains to be well-conditioned and accurate even for zero or small thicknesses. The stability of hybrid matrix method has been demonstrated by the numerical results of reflection and transmission coefficients. These results have been determined efficiently based on the surface hybrid matrix method involving only a subset of hybrid submatrices. In conjunction with the recursive asymptotic method, the hybrid matrix method is self-sufficient without hybrid asymptotic method and may achieve low error level over a wide range of sublayer thickness or the number of recursive operations.

Frictionless contact in a layered piezoelectric medium characterized by complex eigenvalues

Abstract: A local/global stiffness matrix formulation is presented to study the response of an arbitrarily multilayered piezoelectric half-plane indented by a rigid frictionless parabolic punch. The methodology is extended to accommodate the presence of piezoelectric layers that are characterized by complex eigenvalues. Different arrangements of elastic and transversely orthotropic piezoelectric materials within the multilayered medium are considered. A generalized plane deformation is used to obtain the governing equilibrium equations for each individual layer. These equations are solved using the infinite Fourier transform technique. The problem is then reformulated using the local/global stiffness method, in which a local stiffness matrix relating the stresses and electric displacement to the mechanical displacements and electric potential in the transformed domain is formulated for each layer. Then it is assembled into a global stiffness matrix for the entire half-plane by enforcing continuity conditions along the interface. Application of the mixed boundary conditions reduces the problem to an integral equation for the unknown pressure in the contact area. This integral has a divergent kernel that is decomposed into a Cauchy-type and regular parts using the asymptotic properties of the local stiffness matrix. The resulting equation is numerically solved for the unknown contact pressure using a technique based on the Chebyshev polynomials.

Actuation concepts for variable stiffness materials

Abstract: Apparatus and associated methods for actuating variable stiffness material (VSM) structures and achieving deformation of the structures. The apparatus and the associated methods use internal embedded actuation elements and/or externally attached elements to the VSM structures to achieve the desired deformation. In particular, the actuation can be changed due to the variable stiffness nature of the materials. That is, the invention provides the ability to control the deformation of structures using local stiffness control over subregions of the component in addition to or in substitution for actuation. Furthermore, the invention exploits the variable stiffness properties of the VSM structures to enable new functionalities impossible to realize with conventional constant stiffness materials.

History of the stiffness method

Abstract: This paper presents a brief history of the development of the stiffness method. We start by tracing the evolution of the method to solve discrete-type problems such as trusses and frames composed of two node members. We then describe the method as it is applied to solve continuum problems modelled by finite-difference and finite-element methods.

Cellular variable stiffness materials for ultra-large reversible deformations in reconfigurable structures

Abstract: Structures that can physically adapt to fulfill many roles can enable a new generation of high-performance military systems. The key to achieving substantial benefit from shape-changing operations is large changes in structural geometry and stiffness. In this study, we demonstrate variable stiffness cellular materials capable of large global changes in area through local buckling modes. Furthermore, stiffness properties and Poisson ratios may be tailored to provide desirable structural reconfiguration properties such as negative Poisson ratio and highly anisotropic stiffness. However, stiffness properties of cellular materials are two to three orders of magnitude below their constitutive materials properties. Their elastic properties can vary considerably as a function of the applied strain level due to the redistribution of structural material within the cells. Another complication is the difficulty in controlling the local buckling mode due to sensitivity to boundary conditions and loading conditions.

Stiffness and Strength of Multilayer Beams

Abstract: This article describes an original method for calculating stiffness and strength of multilayer beams (MBs) when the beam is deformed within elasticity limits. It draws on previous studies by the same author, but for the first time fully describes the proposed method, both the theory and its practical application. This method is very useful for engineering calculations and scientific research. The equations obtained allow establishing the positions of neutral layers and of the geometric and stiffness centers when the cross section of the beam is not symmetric to both x- and y-axes. This study also presents the equations for establishing the stiffness for bending and shear, and for calculating normal and shear stresses at any point of the beam’s cross section. The efficiency and practical application of this new method are demonstrated by analyzing how MB stiffness and strength change depending on the influence of certain factors, such as mechanical properties of layers, number of layers, and their arrangem...

An Algebraic Multigrid Method for Higher-order Finite Element Discretizations

Abstract: In this paper, we will design and analyze a class of new algebraic multigrid methods for algebraic systems arising from the discretization of second order elliptic boundary value problems by high-order finite element methods. For a given sparse stiffness matrix from a quadratic or cubic Lagrangian finite element discretization, an algebraic approach is carefully designed to recover the stiffness matrix associated with the linear finite element disretization on the same underlying (but nevertheless unknown to the user) finite element grid. With any given classical algebraic multigrid solver for linear finite element stiffness matrix, a corresponding algebraic multigrid method can then be designed for the quadratic or higher order finite element stiffness matrix by combining with a standard smoother for the original system. This method is designed under the assumption that the sparse matrix to be solved is associated with a specific higher order, quadratic for example, finite element discretization on a finite element grid but the geometric data for the underlying grid is unknown. The resulting new algebraic multigrid method is shown, by numerical experiments, to be much more efficient than the classical algebraic multigrid method which is directly applied to the high-order finite element matrix. Some theoretical analysis is also provided for the convergence of the new method.

Continuous and Finite Element Methods for the Vibrations of Inflatable Beams

Abstract: Inflatable structures are under increasing development in various domains. Their study is often carried out by using 3D membrane finite elements and for static loads. There is a lack of knowledge in dynamic conditions, especially for simple and accurate solutions for inflatable beams. This paper deals with the research on the natural frequencies of inflatable Timoshenko beams by an exact method: the continuous element method (CEM), and by the classical finite element method (FEM). The dynamic stiffness matrix D(ω) is here established for an inflatable beam; it depends on the natural frequency and also on the inflation pressure. The stiffness and mass matrixes used in the FEM are deduced from D(ω). Natural frequencies and natural modes of a simply supported beam are computed, and the accuracy of the CEM is checked by comparisons with the finite element method and also with experimental results.

Bending analysis of folded plates by the FSDT meshless method

Abstract: In this paper, a meshfree Galerkin method that is based on the first-order shear deformation theory (FSDT) will be introduced to analyse the elastic bending problem of stiffened and un-stiffened folded plates under different loadings and boundary conditions. Folded plates are regarded as assemblies of plates that lie in different planes. The stiffness matrices of the plates are given by the meshfree method. Employing the element concept, which is borrowed from the finite element method, and treating every plate as a big element, the global stiffness matrix of the whole folded plate is obtained by superposing the stiffness matrices of the plates. This is about the same for the analysis of stiffened folded plates. They are considered as assemblies of stiffened plates. The stiffness matrices of the stiffened plates are also given by the meshfree method. Superior to the finite element methods, no mesh is required in determining the stiffness matrices for the plates and the stiffened plates in this paper, which means time-consuming and accuracy-suffering remeshing is entirely avoided for problems such as large deformation or crack propagation in folded plates or stiffener position changes of stiffened folded plates. To demonstrate the accuracy and convergence of the method, several numerical examples are calculated by it and the finite element commercial software ANSYS. Good agreement is observed between the two sets of results.

The analytical element stiffness matrix of a recent 4‐node membrane element formulated by the quadrilateral area co‐ordinate method

Abstract: The quadrilateral area co-ordinate (QAC) method is a new tool for developing quadrilateral finite element models. Compared with those traditional elements using isoparametric co-ordinates, models formulated by QAC method are less sensitive to mesh distortion. Furthermore, another advantage for QAC system is that the analytical expressions for element stiffness matrix can be obtained theoretically by basic formulae of QAC integrals, which are beneficial to finite element computation procedure. However, because of the relative complexity of the derivation process, no such explicit form of the stiffness matrix has been presented before. In this paper, the analytical element stiffness matrix of a recent 4-node quadrilateral membrane element, AGQ6-I, is given out for the first time. This element AGQ6-I, which was constructed by QAC method and generalized conforming conditions, successfully overcomes various locking problems and exhibits much better performances than many isoparametric models. And its formulations have also been introduced in shell analysis by other researchers. Numerical examples show that obvious higher computation efficiency can be achieved by presented explicit formulations. So the analytical form of the element stiffness matrix of element AGQ6-I possesses significance for its further applications. Copyright © 2006 John Wiley & Sons, Ltd.

Stiffness Modeling for an Orthogonal 3-PUU Compliant Parallel Micromanipulator

Abstract: The stiffness modeling for a compliant parallel manipulator (CPM) is very important since it provides a basis for the characterization of static, modal, and dynamic behavior of the CPM. This paper presents the stiffness modeling of a three-prismatic-universal-universal (3-PUU) CPM with orthogonally mounted actuators, that is designed to provide three spatial translational DOF for nano-scale manipulation. Considering the compliance of each compliant element, the analytical stiffness model for a spatial CPM is established by a straightforward approach, which is then applied to stiffness modeling of the 3-PUU CPM. In addition, the finite element analysis is carried out to validate the developed model, and as a further application, the influence of architectural parameters on stiffness factors are derived based on the stiffness model, which is valuable for a cost-effective design of the CPM.

Semi‐analytical integration of the 8‐node plane element stiffness matrix using symbolic computation

Abstract: The semi-analytical integration of an 8-node plane strain finite element stiffness matrix is presented in this work. The element is assumed to be super-parametric, having straight sides. Before carrying out the integration, the integral expressions are classified into several groups, thus avoiding duplication of calculations. Symbolic manipulation and integration is used to obtain the basic formulae to evaluate the stiffness matrix. Then, the resulting expressions are postprocessed, optimized, and simplified in order to reduce the computation time. Maple symbolic-manipulation software was used to generate the closed expressions and to develop the corresponding Fortran code. Comparisons between semi-analytical integration and numerical integration were made. It was demonstrated that semi-analytical integration required less CPU time than conventional numerical integration (using Gaussian-Legendre quadrature) to obtain the stiffness matrix. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006

GA-Based Architecture Optimization of a 3-PUU Parallel Manipulator for Stiffness Performance

Abstract: In this paper, an optimal design of a three-prismatic-universal-universal (3-PUU) translational parallel manipulator is performed using the genetic algorithm (GA) to achieve optimum stiffness characteristics. The stiffness matrix is derived intuitively based upon an overall Jacobian with the consideration of actuation and constraints, and the compliance subjected to both actuators and legs is taken into account to establish the stiffness model. A global stiffness index defined as the mean value of the determinant of stiffness matrix over the reachable workspace is adopted as a stiffness measure, meanwhile the architectural parameters are optimized via a GA approach. The research results are valuable in the design of a 3-PUU parallel manipulator for machine tool applications.

RC large displacements: Optimization applied to experimental results

Abstract: A large number of towers for telecommunication were installed during the implantation of cellular telephony services in Brazil. Some of those towers presented problems as excessive displacements, residual displacements, cracking and some accidents happened. On the other hand, the computation of large displacements in slender reinforced concrete structures is a very difficult task as the flexural stiffness of the sections changes continuously as the bending moment increases, due to the very non-linear material behavior of concrete, involving such phenomena as formation of cracks and plastification. The goal of this paper is to present some initial results of the application of optimization techniques to experimental data relative to the determination of the effective bending stiffness of transverse sections of reinforced concrete structures. The objective is to determine parameters of reduction of the stiffness of unstressed sections for the correct calculation of the displacements of those structures. The results of a test with a reinforced concrete tower for telecommunications of 30m of length, circular ring cross-section with 50cm diameter, were used. For several cross-sections along the axis of this structure the effective stiffness was computed. For analysis purposes, the structure was discretized and the differential equation of the elastic line integrated to obtain the rotations and displacements. The values of the effective stiffness of the cross-sections were obtained using optimization techniques. The effective stiffness is presented in graphs as function of the solicitation level (the ratio between the characteristic bending moment and the ultimate moment of the cross-section). The section where the largest stiffness loss happened is the section that indeed collapsed in a real similar structure. Directions for future researches are presented.

Numerical structural analysis system based on the load-transfer-path method

Abstract: The purpose of this invention is to reduce the calculation time in the numerical structure analysis system based on load-transfer-path method. The parameters are set in the condition that the supporting point B in the objective structure is fixed and the load is applied to the specific loading point A. The FEM calculation means 2 calculates the deformation of the objective structure according to the structural stiffness matrix in the stiffness matrix holding means 1 to find the basic data such as the displacement of each point and so on. The FEM calculation means calculates each deformation to find the displacement under the condition that the specific loading point A and the supporting point B are fixed and three inspection loadings are applied to the variable loading point C. The partial stiffness matrix calculation means 3 solves the multidimensional simultaneous linear equation based upon the internal stiffness matrix of the objective structure, the load value and the displacement to find the partial stiffness matrix K AC . The stiffness parameter calculation means 8 calculates the value of the stiffness parameter U* according to the partial stiffness matrix K AC and the displacement in the basic data and so on. The value of U* of each point is calculated with changing the variable loading point C as to follow sequentially all the necessary points in the objective structure.

Stiffness of a Flexible Circular Footing Embedded in an Elastic Half-Space

Abstract: This paper addresses the problem of a circular footing of finite but nonzero stiffness embedded in an isotropic nonhomogeneous elastic half-space. The problem is solved by coupling the scaled boundary finite-element method with axisymmetric shell finite elements. The coupling of the two methods is validated by comparing computed solutions with analytical solutions for a flexible footing subjected to a uniformly distributed vertical load embedded in an elastic full-space. Based on the finding that the bending stiffness of the footing dictates the response for vertical and moment load cases, whereas the normal stiffness of the footing dominates the horizontal response, a convenient method for estimating dimensionless elastic stiffness coefficients is presented graphically. New results are presented for homogeneous and Gibson soil profiles and Poisson’s ratios of 0.2 and 0.499, to represent both sand and clay. An example demonstrating a practical application of these results is also provided.

Dynamic stiffness for thin-walled structures by power series

Abstract: The dynamic stiffness method is introduced to analyze thin-walled structures including thin-walled straight beams and spatial twisted helix beam. A dynamic stiffness matrix is formed by using frequency dependent shape functions which are exact solutions of the governing differential equations. With the obtained thin-walled beam dynamic stiffness matrices, the thin-walled frame dynamic stiffness matrix can also be formulated by satisfying the required displacements compatibility and forces equilibrium, a method which is similar to the finite element method (FEM). Then the thin-walled structure natural frequencies can be found by equating the determinant of the system dynamic stiffness matrix to zero. By this way, just one element and several elements can exactly predict many modes of a thin-walled beam and a spatial thin-walled frame, respectively. Several cases are studied and the results are compared with the existing solutions of other methods. The natural frequencies and buckling loads of these thin-walled structures are computed.

Fourier series based finite element analysis of tube hydroforming: An axisymmetric model

Abstract: Purpose – The purpose of this paper is to provide an axisymmetric model of tube hydroforming using a Fourier Series based finite element method.Design/methodology/approach – Fourier series interpolation function, which considerably reduces the size of the global stiffness matrix and the number of variables, is employed to approximate displacements. The material of the tube is assumed to be elastic‐plastic and to satisfy the plasticity model that takes into account the rate independent work hardening and normal anisotropy. Numerical solution obtained from an updated Lagrangian formulation of the general shell theory is employed. The axial displacement stroke (a.k.a. axial feed) during tube hydroforming is incorporated using Lagrange multipliers. Contact constraints and boundary friction condition are introduced into the formulation based on the penalty function, which imposes the constraints directly into the tangent stiffness matrix. A forming limit curve based on shear instability and experimental measur...