Performance of Walsh-based hybrid-mixed stress analysis

The most relevant aspects involved in the implementation of the stress model of the hybrid-mixed finite element formulation are presented and discussed. In this formulation, the stress and displacement fields in the domain of the element and the displacements on the boundary are simultaneously approximated. Digital Walsh functions are used in all three approximations. Due to the special properties of these functions, it is possible to obtain closed form solutions for the integrals involved in the computation of the structural operators. The governing system is usually very large in dimension, but also well structured and highly sparse. Consequently, the use of adequate algorithms to store and manipulate sparse matrices is essential to ensure the numerical efficiency of the model. The direct and iterative methods most used in the solution of large, sparse systems of linear equations are tested and assessed. Fast transform algorithms are used in the post-processing phase to perform the linear combinations of digital Walsh functions needed to construct the stress and displacement fields from their direct approximations. Linear elastostatic problems are used to illustrate the implementation of the Walsh-based hybrid-mixed stress elements.     

Analysis of continuum by the integrated force method

The novel formulation termed the integrated force method (IFM) has been established for finite element discrete analysis. In this paper we have extended the IFM for the analysis of continuum taking circular plate as the example. The primary variables of the analysis are moments. All the continuum equations (equilibrium equations and compatibility conditions) in the field and on the boundary are obtained in moments from the stationary condition of the variational functional of the IFM. A new stress function required for the functional is defined. The variational functional yields the known equations along with the novel boundary condition identified as the boundary compatibility condition. The moment solution for the plate problem is obtained without any recourse to displacements either in the field or on the boundary. From moments, displacements are obtained by integration and boundary displacement continuity conditions. The IFM solution and boundary compatibility conditions are verified using Timoshenko's work and finite element displacement method.     

A coupled formulation of finite and boundary element methods in two-dimensional elasticity

A symmetric stiffness formulation based on a boundary element method is studied for the structural analysis of a shear wall, with or without cutouts. To satisfy compatibility requirements with finite beam elements and to avoid problems due to the eventual discontinuities of the traction vector, different interpolation schemes are adopted to approximate the boundary displacements and tractions. A set of boundary integral equations is obtained with the collocation points on the boundary, which are selected by the error minimization technique proposed in this paper, and the stiffness matrix is formulated with those equations and symmetric coupling techniques of finite and boundary element methods. The newly developed plane stress element can have the openings in its interior domain and can be easily linked with the finite beam/column elements.     

T-elements: a finite element approach with advantages of boundary solution methods

Since their introduction in 1977, the so-called T-elements have considerably evolved and have now become a highly efficient and well established tool for the solution of complex boundary value problems. This class of finite elements, associated with the Trefftz method, is based on enforcement of interelement continuity and boundary conditions on assumed displacement fields chosen so as to a priori satisfy the governing differential equations of the problem. Several alternative T-element formulations are available which yield for a particular subdomain the customary force-displacement relationship with a symmetric positive definite stiffness matrix which makes it possible for such elements to be implemented into the standard finite element (FE) codes.

Owing to their nature, the T-elements may either be considered as a new FE model or as a non-conventional symmetric substructure-oriented form of the boundary element method (BEM). From the point of view of the latter, the outstanding features of the T-element approaches are the use of T-complete sets of non-singular solutions (rather than the singular Kelvin's type fundamental solutions) and the replacement of the customary integral equations form by a symmetric variational formulation.

This paper reviews and critically assesses the most important T-element formulations developed over the past years. It shows that such elements not only cumulate the advantages and discard the drawbacks of the conventional finite element and boundary element methods, but also offer additional advantages not available in the standard form of these methods.     

Quadratic divergence-free finite elements on Powell–Sabin tetrahedral grids

Given a tetrahedral grid in 3D, a Powell–Sabin grid can be constructed by refining each original tetrahedron into 12 subtetrahedra. A new divergence-free finite element on 3D Powell–Sabin grids is constructed for Stokes equations, where the velocity is approximated by continuous piecewise quadratic polynomials while the pressure is approximated by discontinuous piecewise linear polynomials on the same grid. To be precise, the finite element space for the pressure is exactly the divergence of the corresponding space for the velocity. Therefore, the resulting finite element solution for the velocity is pointwise divergence-free, including the inter-element boundary. By establishing the inf-sup condition, the finite element is stable and of the optimal order. Numerical tests are provided.     

Mixed finite element formulations of strain-gradient elasticity problems

Theories on intrinsic or material length scales find applications in the modeling of size-dependent phenomena. In elasticity, length scales enter the constitutive equations through the elastic strain energy function, which, in this case, depends not only on the strain tensor but also on gradients of the rotation and strain tensors. In the present paper, the strain-gradient elasticity theories developed by Mindlin and co-workers in the 1960s are treated in detail. In such theories, when the problem is formulated in terms of displacements, the governing partial differential equation is of fourth order. If traditional finite elements are used for the numerical solution of such problems, then C¹ displacement continuity is required. An alternative “mixed” finite element formulation is developed, in which both the displacement and the displacement gradients are used as independent unknowns and their relationship is enforced in an“integral-sense”. A variational formulation is developed which can be used for both linear and non-linear strain-gradient elasticity theories. The resulting finite elements require only C⁰ continuity and are simple to formulate. The proposed technique is applied to a number of problems and comparisons with available exact solutions are made.     

Reduced degree displacement functions for instability and nonlinear structural analysis

A procedure for simplifying the derivation of stiffness matrices used in the finite element analysis of instability and nonlinear structural problems is presented. The displacement functions assumed to derive the nonlinear components of element stiffness matrices provide inter-element continuity of displacement derivatives of order one less than appear in the energy functional and therefore comply with established convergence criteria for finite element analysis. For the class of problems discussed, this implies the use of linear polynomial displacement functions, which simplifies the derivation considerably and avoids the need for complex numerical integration. A number of practical examples are discussed to illustrate the effectiveness of this procedure.     

Finite element method in plane Cosserat elasticity

A displacement and rotation based finite element method for the solution of boundary value problems in linear isotropic Cosserat elasticity is proposed. The field equations for the problem of plane strain are derived from the three-dimensional theory and expressed in oblique rectilinear coordinates. Three fundamental triangular finite elements are presented. A patch test for validating Cosserat finite element formulations is also introduced.     

Shape design sensitivity analysis of kinematical boundaries

A new approach is used in this paper to derive the design sensitivity formulation with kinematical design boundaries. By employing the concept of the conventional finite difference approach, the variation of structural response due to change of the kinematic design boundary can be represented by the perturbed structure under a set of kinematical boundary conditions. Parameterization of the design variation with respect to the design variable enables us to transform the design sensitivity into the solutions of a boundary value problem with perturbation displacements on the design boundary. The perturbation diplacements can be evaluated from the stress and displacement fields of the initial problem. This approach can be treated as a special case of the general direct formulation, but the derivation using the finite difference procedure gives a strong physical meaning of the method, and the formulation derived provides an explicit form for design sensitivity calculation. The numerical implementation of this approach based on the boundary element method is discussed, and a few numerical examples are used to verify the proposed formulation.     

Dynamic finite element analysis of nonaxisymmetric structures

A dynamic finite element method of analysis is developed for structural configurations which are derived from axisymmetric geometries but contain definite nonaxisymmetric features in the circumferential direction. The purpose of the analysis is to develop a method which will take into consideration the fact that the stress and strain conditions in these geometries will be related to the corresponding axisymmetrie solution. This analysis is an extension of previously published work in which a similar approach was developed for static structural problems. The analysis is developed in terms of a cylindrical coordinate system r, θ and z. As the first step of the analysis, the geometry is divided into several segments in the r-θ plane. Each segment is then divided into a set of quadrilateral elements in the r-z plane. The axisymmetric displacements are obtained for each segment by solving a related axisymmetric configuration. A perturbation analysis is then performed to match the solutions at certain points between the segments, and obtain the perturbation displacements for the total structure. The total displacement is then the axisymmetric displacement plus the perturbation displacement. The analysis allows for elastic-plastic materials with orthotropic yield criterion based on Hill's yield function and kinematic strain hardening. The finite element dynamic equations are solved by finite differences by dividing the time domain into incremental steps. The solution has been programmed on a computer and applied to a number of examples.     

Effect of bolt load on the deformation of a taper hub flange

This paper presents an investigation to study the effect of bolt load on the displacement and stress pattern of a taper hub flange. The flange is analysed both by finite difference and finite element methods. Novozhilov's self-consistent shell theory is applied in deriving the equations of equilibrium in the three dimensions. The axial displacements are important in any flange design with regard to the leakage characteristics of such a joint hence the preference to the above theory. The tapered hub is approximated by a series of consecutive stepped cylindrical shells. While applying the finite difference method, the flange ring is suitably modelled using the concept of branching of shells. In this work one such flange is analysed and the results of finite difference and finite element methods are found to be in agreement.     

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.     

Structural optimization using global stress-deviation objective function via the level-set method

The paper deals with minimum stress design using a novel stress-related objective function based on the global stress-deviation measure. The shape derivative, representing the shape sensitivity analysis of the structure domain, is determined for the generalized form of the global stress-related objective function. The optimization procedure is based on the domain boundary evolution via the level-set method. The elasticity equations are, instead of using the usual ersatz material approach, solved by the extended finite element method. The Hamilton-Jacobi equation is solved using the streamline diffusion finite element method. The use of finite element based methods allows a unified numerical approach with only one numerical framework for the mechanical problem as also for the boundary evolution stage. The numerical examples for the L-beam benchmark and the notched beam are given. The results of the structural optimization problem, in terms of maximum von Mises stress corresponding to the obtained optimal shapes, are compared for the commonly used global stress measure and the novel global stress-deviation measure, used as the stress-related objective functions.     

Mixed finite element models for plate bending analysis theory

The theoretical background of mixed finite element models, in general for nonlinear problems, is briefly reexamined. In the first part of the paper, several alternative “mixed” formulations for 3-D continua undergoing large elastic deformations under the action of time dependent external loading are outlined and are examined incisively. It is concluded that mixed finite element formulations, wherein the interpolants for the stress field satisfy only a part of the domain equilibrium equations, are not only consistent from a theoretical standpoint but are also preferable from an implementation point of view. In the second part of the paper, alternative variational bases for the development of thin-plate elements are presented and discussed in detail. In light of this discussion, it is concluded that the “bad press” generated in the past concerning the practical relevance of the so-called assumed stress hybrid finite element model is not justified. Moreover, the advantages of this type of elements as compared with the “assumed displacement” or alternative mixed elements are outlined.     

Fracture analysis of plane piezoelectric/piezomagnetic multiphase composites under transient loading

The transient response of cracked composite materials made of piezoelectric and piezomagnetic phases, when subjected to in-plane magneto-electro-mechanical dynamic loads, is addressed in this paper by means of a mixed boundary element method (BEM) approach. Both the displacement and traction boundary integral equations (BIEs) are used to develop a single-domain formulation. The convolution integrals arising in the time-domain BEM are numerically computed by Lubich’s quadrature, which determines the integration weights from the Laplace transformed fundamental solution and a linear multistep method. The required Laplace-domain fundamental solution is derived by means of the Radon transform in the form of line integrals over a unit circumference. The singular and hypersingular BIEs are numerically evaluated in a precise and efficient manner by a regularization procedure based on a simple change of variable, as previously proposed by the authors for statics. Discontinuous quarter-point elements are used to properly capture the behavior of the extended crack opening displacements (ECOD) around the crack-tip and directly evaluate the field intensity factors (stress, electric displacement and magnetic induction intensity factors) from the computed nodal data. Numerical results are obtained to validate the formulation and illustrate its capabilities. The effect of the combined application of electric, magnetic and mechanical loads on the dynamic field intensity factors is analyzed in detail for several crack configurations under impact loading.     

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.     

A robust non-linear solid shell element based on a mixed variational formulation

The paper is concerned with a geometrically non-linear solid shell finite element formulation, which is based on the Hu-Washizu variational principle. For the approximation of the independent displacement, stress and strain fields, the strain field is additively decomposed into two parts. Due to the fact that one part of the strain field is interpolated in the same manner as proposed by the enhanced assumed strain (EAS) method, it is denoted as EAS field. The other strain field is approximated with the same interpolation functions as the stress field. In contrast to the EAS concept the approximation spaces of the stresses and the enhanced assumed strains are not orthogonal. Consequently the stress field is not eliminated from the finite element equations. For the displacements tri-linear shape functions are considered. Shear locking and curvature thickness locking are treated using assumed natural strain interpolations. A static condensation leads to a simple low order hexahedral solid shell element. Numerical tests show that the present model is very robust and allows larger load steps than an EAS solid shell element.     

The method of lines applied to crack problems including plasticity effect

To seek stress and strain distributions in flawed structures, the governing equations in elasto-plasticity are first derived in terms of displacement increments. The method of lines is then applied to derive the sets of ordinary differential equations with appropriate boundary conditions. Their solutions are sought along continuous lines of a discretized region. They are solved by a combination of power series and modal matrix method. A step by step integration is devised to determine the displacements at nodal points. Non-linear work-hardening is taken care of by the effective stress and strain approach. The cases of uncracked and cracked hollow cylinders of finite length under axisymmetrical loadings are studied in detail. The growth of plastic zone, crack-opening displacement and J-integral along various paths are illustrated.     

A special BEM for elastostatic analysis of building floor slabs on columns

This work presents a boundary element formulation for the analysis of building floor slabs, without beams, in which columns are coupled with the plate. An alternative formulation of boundary element method is presented, which considers three nodal displacements values (w, ∂w/∂n and ∂w/∂s) for the nodes at the boundary of the plate. In this formulation three boundary equations are written for all nodes at the boundary and in the domain of the plate. As the nodes of the column-plate connections are also represented by three nodal values, all these structural elements can be easily coupled. It is supposed that the cross-sections of the columns remain flat after the deflection and consequently the assumption of linear variation of the stress in the plate-column contact surface is also valid.