Igor Kaljević

Development of Finite Elements for Two-Dimensional Structural Analysis Using the Integrated Force Method

The integrated force method has been developed in recent years for the analysis of structural mechanics problems. In the intgrated force method all independent forces are treated as unknown variables, which are calculated by simultaneously imposing equations of equilibrium and compatibility conditions. The development of a finite element library for the analysis of two-dimensional problems using the integrated force method is presented in this paper. Elements of triangular and quadrilateral shapes, capable of modeling arbitrary domain configurations are developed. The element equilibrium and flexibility matrices are derived by discretizing expressions for corresponding potential and complementary energies, respectively. Independent approximations of displacement and stress fields within finite elements are performed. Interpolation of the displacement field is done similarly as in the standard displacement method. The stress field is approximated using full polynomials of correct orders. A procedure for deriving the stress interpolation polynomials that utilizes the definitions of stress components in terms of Airys stress function is developed. Such derived stress fields identically satisfy equations of equilibrium, and the resulting element matrices are insensitive to the orientation of local coordinate systems. A method to calculate the number of rigid body modes is devised, and it is shown that the present elements do not possess spurious zero energy modes. A number of example problems are solved using the present library and the results are compared with corresponding analytical solutions and those obtained from the standard displacement finite element method. A good agreement of the results, and better performance of the integrated force method, compared to the displacement method, in stress calculations, is observed.

Three-dimensional structural analysis by the integrated force method

Abstract The “integrated force method”, which has been recently developed for analyzing structures, is extended in this paper for three-dimensional structural analysis. A general formulation to generate the stress interpolation matrix in terms of complete polynomials of the required order is developed first. The formulation is based on the definitions of stress tensor components in terms of stress functions. The stress functions are written as complete polynomials and substituted into expressions for stress components. After eliminating dependent coefficients, the expressions for stress components are obtained as complete polynomials, where coefficients are defined as generalized independent forces. Such derived components of the stress tensor identically satisfy Naviers equations of equilibrium. The resulting element matrices are invariant with respect to coordinate transformation and are free of spurious zero energy modes. The formulation provides a rational way to calculate the exact number of independent forces for the required order of approximation with complete polynomials. The reduction in the number of independent forces and its influence on the accuracy of the response are also analyzed. The stress fields derived here are next used to develop a comprehensive finite element library for the three-dimensional structural analysis using the “integrated force method”. Elements of both tetrahedral and hexahedral shapes, capable of modeling arbitrary geometric configurations, are developed. A number of example problems with available analytical solutions are solved using the present developments and a good agreement of results with the analytical solutions is observed. The responses obtained using the “integrated force method” are also compared with those generated with the standard displacement method. A better overall performance of the “integrated force method” is observed.

Stochastic boundary elements in elastostatics

Abstract A stochastic boundary element formulation for the treatment of boundary value problems in two-dimensional elastostatics that involve a random operator is presented. A general perturbation procedure is formulated for the set of correlated random variables governing the response of the solid. This procedure is then specialized for the cases of (a) random geometry and (b) random material properties. The problems involving a random configuration are analyzed using the random variable model, and those with a random material property are analyzed using the random field model. The random field is first discretized into a set of correlated random variables, which are then transformed into an uncorrelated set to simplify the analysis. The derivatives of the boundary element matrices appearing in the systems of equations from the perturbation of the random variables are derived analytically. The direct solution methods are used to obtain the response variables and their first- and second-order derivatives, respectively. Quadratic, conforming boundary elements are employed in the boundary element discretization and the strongly singular terms of the boundary element matrices and their first- and second-order derivatives are obtained using the conditions associated with rigid body motions of the solid. The present formulation has been evaluated for a number of example problems through comparisons with the solutions obtained by Monte Carlo simulation. A good agreement of the results is observed.

Stochastic boundary elements for two-dimensional potential flow in homogeneous domains

Abstract A stochastic boundary element formulation is presented for the analysis of two-dimensional steady state potential flow through homogeneous domains. The operator of the governing differential equation is assumed to be random and is described by a set of correlated random variables. The perturbation method, in conjunction with the boundary element method, is employed to derive the systems of equations for the unknown boundary variables and their respective first and second order derivatives with respect to the random variables. These quantities are then used to calculate the desired response statistics. A general procedure is developed which is next applied for the specific cases of random geometric configuration and random material parameter. The random geometric configuration is modeled using a finite set of correlated random variables. The random material parameter is modeled as a homogeneous random field which allows the use of deterministic fundamental solutions and integral representations for homogeneous domains. The random field is first discretized into a set of correlated random variables and then the general procedure is applied. A transformation of the correlated random variables into an uncorrelated set is performed to reduce the number of numerical operations. The results for the boundary variables are used to calculate the response statistics of internal potentials. These calculations require the modeling of the interior of the domain under consideration. Several models for representing the interior of the domain are presented for both random configuration and random material parameter and their influence on the response statistics is analysed. Distributed sources are considered in the present study using the particular integral approach. A number of numerical examples are presented to demonstrate the validity of the present formulations. The results obtained from the present analyses are compared with those obtained from Monte Carlo simulations with 5000 samples and a good agreement of results is observed.

Completed Beltrami-Michell formulation for analyzing mixed boundary value problems in elasticity

In elasticity, the method of forces, wherein stress parameters are considered as the primary unknowns, is known as the Beltrami-Michell formulation. The Beltrami-Michell formulation can only solve stress boundary value problems; it cannot handle the more prevalent displacement or mixed boundary value problems of elastidty. Therefore, this formulation, which has restricted application, could not become a true alternative to the Navier displacement method, which can solve all three types of boundary value problems. The restrictions of the Beltrami-Michell formulation have been alleviated by augmenting the classical formulation with a novel set of conditions identified as the boundary compatibility conditions. This new method, which completes the classical force formulation, has been termed the completed Beltrami-Michell formulation. The completed Beltrami-Michell formulation can solve general elasticity problems, with stress, displacement, and mixed boundary conditions in terms of stresses as the primary unknowns. The completed Beltrami-Michell formulation is derived from the stationary condition of the variational functional of the integrated force method. In the completed Beltrami-Michell formulation, stresses for kinematically stable structures can be obtained without any reference to displacements either in the field or on the boundary. This paper presents the completed Beltrami-Michell formulation and its derivation from the variational functional of the integrated force method. Examples are presented to demonstrate the applicability of the completed formulation for analyzing mixed boundary value problems under thermomechanical loads. Selected examples include analysis of a composite cylindrical shell, wherein membrane and bending response are coupled, and a composite circular plate.

Stochastic boundary elements for two-dimensional potential flow in non-homogeneous media

Abstract A stochastic boundary element formulation is presented for the treatment of two-dimensional problems of steady-state potential flow in non-homogeneous media that involve a random operator in the governing differential equation. The randomness is introduced through the material parameter of the domain which is described as a non-homogeneous random field. The random field is discretized into a set of correlated random variables and a perturbation is applied to the differential equation of the problem. This leads to differential equations for the unknown potential and its first- and second-order derivatives, respectively, evaluated at the mathematical expectations of the random variables resulting from the discretization of the random field. An approximate method is applied for the solution of these equations by expressing the potential and its derivatives as a sum of functions of descending order. These solutions are introduced into the differential equations and upon equating similar order terms, a sequence of Poissons equations is obtained for each order. A transformation of the correlated random variables into an uncorrelated set is performed to reduce the number of numerical operations by retaining a small number of transformed random variables. The resulting equations are solved using the boundary element method to obtain the unknown boundary values of the potentials and their respective first- and second-order derivatives which are then used to compute the desired response statistics. Quadratic, conforming boundary elements are used in the boundary integration and four-node quadrilateral cells are used in the domain integration. Strongly singular terms of the boundary element matrices are obtained indirectly by applying a state of uniform unit potential over the entire contour of the object. The singular domain integrals are calculated analytically. Direct solution techniques are used to calculate the response variables and their derivatives, respectively. A number of example problems are presented and the results are compared with those obtained from Monte Carlo simulations. A good agreement of the results is observed.

Treatment of initial deformations in the Integrated Force Method

Abstract A new force finite element formulation, termed the Integrated Force Method, has been developed in recent years for the structural analysis. In the Integrated Force Method all independent forces are treated as unknown variables which are calculated by simultaneously imposing the force equilibrium and strain compatibility. The Integrated Force Method is extended in this study for the treatment of initial strains. Initial strains are incorporated into compatibility conditions through the vector of initial deformations. A general expression for the vector of initial deformations is derived by discretizing the expression for the complementary potential energy of the structure. Two special cases of initial strains are analyzed in detail: (i) thermal strains and (ii) support settlements. The vector of initial deformations due to thermal loadings is derived by introducing expressions for thermal strains into the general expression, and that due to support settlements is derived directly by equating the work done by external and internal forces, respectively. Two example problems with available analytical solutions are solved using the present developments. A good agreement of results with analytical solutions is observed.

Random Interior Data Representations in Probabilistic Boundary Element Anallysis

Stochastic boundary element formulations have been recently developed for analyzing boundary value problems described by differential equations with random operators [1,2]. The domain excitations in these formulations were treated using the particular integral approach [3] wherein random properties at locations within the domain were expressed in terms of their values on the boundary only. The response on the boundary was independent of the variation of random properties inside the domain, and that at an internal point was affected by the random properties at that location only. A careful examination of the corresponding integral representations reveals that, in general, response statistics for locations on both the contour and within the domain, may depend on random properties of the entire object.