Showing papers in "Computational Mechanics in 1989"

Nonlinear bending analysis of orthotropic rectangular plates by the method of differential quadrature

Abstract: The behavior of thin, rectangular, orthotropic elastic plates, with immovable edges and undergoing large deflections, is investigated by the numerical technique of differential quadrature. Approximate results are obtained, using the Newton-Raphson method and, alternatively, a finite-difference-based method to solve the nonlinear systems of equations. Bending stresses, membrane stresses, and deflections are calculated for plates with fully clamped and simply supported flexural edge conditions under uniform pressure loading. Results are compared with existing analytical, numerical, and experimental ones. The present method gives good accuracy and is computationally efficient.

Optimality criteria: a basis for multidisciplinary design optimization

Abstract: This paper presents a generalization of what is frequently referred to in the literature as the optimality criteria approach in structural optimization. This generalization includes a unified presentation of the optimality conditions, the Lagrangian multipliers, and the resizing and scaling algorithms in terms of the sensitivity derivatives of the constraint and objective functions. The by-product of this generalization is the derivation of a set of simple nondimensional parameters which provides significant insight into the behavior of the structure as well as the optimization algorithm. A number of important issues, such as, active and passive variables, constraints and three types of linking are discussed in the context of the present derivation of the optimality criteria approach. The formulation as presented in this paper brings multidisciplinary optimization within the purview of this extremely efficient optimality criteria approach.

A regularized boundary integral equation method for elastodynamic crack problems

Abstract: This paper presents a double layer potential approach of elastodynamic BIE crack analysis. Our method regularizes the conventional strongly singular expressions for the traction of double layer potential into forms including integrable kernels and 0th, 1st and 2nd order derivatives of the double layer density. The manipulation is systematized by the use of the stress function representation of the differentiated double layer kernel functions. This regularization, together with the use of B-spline functions, is shown to provide accurate numerical methods of crack analysis in 3D time harmonic elastodynamics.

On continuum damage-elastoplasticity at finite strains

Abstract: A strain-based continuum damage-elastoplasticity formulation at finite strains is proposed based on an additive split of thestress tensor. Within the proposed framework, a hyperelastic extension of the classicalJ 2-flow theory is developed as a model problem, with a rate-free formulation of the (linear) kinematic hardening law that is free from spurious stress oscillation in the simple shear test. The algorithmic implementation of the coupled damage-elastoplasticity model is shown to reduce to a trivial modification of the classical radial return which is amenable toexact linearization. This results in a closed form expression for theconsistent elastoplastic-damage modulus. The algorithmic treatment of the damage model with no restrictions on the functional forms governing the plastic response is considered subsequently. It is emphasized that objective rates and incrementally objective algorithms play no role in the present approach. A number of numerical experiments are presented that illustrate the performance of the proposed formulation.

Non-hyper-singular integral-representations for velocity (displacement) gradients in elastic/plastic solids (small or finite deformations)

Abstract: New integral representations for deformation (velocity) gradients in elastic or elasto-plastic solids undergoing small or large deformations are presented. Compared to the cases wherein direct differentiation of the integral representations for displacements (or velocities) were carried out to obtain displacement (or velocity) gradients (which gave rise to hyper-singularity when the source point was taken to the boundary), the present integral representations have lower order singularities which are quite tractable from a numerical point of view. Moreover, the present representations, allow the source point to be taken in the limit, to the boundary, without any difficulties. This obviates a need for a two tier system of evaluation of deformation gradients in the interior of the domain on one hand and at the boundary of the domain on the other. It is expected that the present formulations would yield more accurate and stable deformation gradients in problems dominated by geometric and material nonlinearities.

High-accuracy bench mark solutions to natural convection in a square cavity

Abstract: A fourth-order high-accuracy finite difference method is presented for the bouyancy-driven flow in a square cavity with differentially heated vertical walls. The two bench mark solutions against which other solutions can be compared were obtained. The present solution is seemed to be accurate up to fifth decimal. The proposed scheme is stable and convergent for high Rayleigh number, and will be applicable to general problems involving flow and heat transfer, especially in three dimensions.

Simultaneous nonlinear structural analysis and design

Abstract: Optimization techniques are increasingly being used for performing nonlinear structural analysis. Under these circumstances the structural design problem can be viewed as a nested optimization problem. The present paper suggests that there are computational benefits to treating this nested problem as a large single optimization problem. That is, the response variables (such as displacements) and the structural parameters are all treated as design variables in a unified formulation which performs simultaneously the design and analysis. Three truss examples are used for the demonstration comparing two nested optimization procedures with two computational procedures for the simultaneous solution. The examples show that the simultaneous approach is competitive with the more traditional nested approach.

η method of numerical integration

Abstract: In the finite element dynamic analysis, the governing partial differential equations are first discretized in space and then the resulting equations are integrated with respect to time. The time integration is an important aspect of the entire analysis since efficiency, economy and accuracy of the solution depends on it, to a large extent. In this paper, a new one step implicit algorithm known as ‘ν method’ suitable for wave propagation problems is introduced. The proposed algorithm includes a term defining an impulse load vector which permits the use of time increments that can be controlled solely by accuracy requirements. The stability and accuracy characteristics of the proposed method are compared with those of the other available methods.

On the Formulation of Variational Theorems Involving Volume Constraints

Abstract: A continued concern with variational theorems which are suitable for numerical implementation in connection with the analysis of incompressible or nearly incompressible materials has led us to the formulation of five-field theorems for displacements, deviatoric stresses, pressure, distortional strains and volume change. In essence these theorems may be thought of as generalizations of the Hu-Washizu three-field theorem for displacements, stresses and strains and of earlier two-field theorems for displacements and stresses.

Boundary-integral equation method for elastodynamic scattering by a compact inhomogeneity

Abstract: The set of singular integral equations which relates unknown fields on the surface of the scatterer to a time-harmonic incident wave is solved by the boundary element method. The general method of solution is discussed in some detail for scattering by an inclusion. Results are presented for a spherical cavity, and for a soft and a stiff spherical inclusion. Fields on the surface of the scatterer are compared with previous results obtained by different methods. Back-scattered and forward-scattered displacement fields are presented, both as a function of position at fixed frequency, and as a function of frequency at fixed position. The quasi-static approximation is briefly discussed.

Boundary integral equation for wave equation with moving boundary and applications to compressible potential aerodynamics of airplanes and helicopters

Abstract: This work presents a general boundary-integral-equation methodology for the solution of the wave equation around objects moving in arbitrary motion, with applications to compressible potential aerodynamics of airplanes and helicopters. The paper includes the derivation of the boundary integral equation for the wave equation, in a frame of reference moving in arbitrary motion (in particular, in translation and in rotation). The formulation is then applied to study unsteady potential compressible aerodynamic flows around streamlined bodies, such as airplanes and helicopters. The formulation is given in terms of the velocity potential, for which an explicit treatment of the wake is required; a discussion of the formulation for the wake transport is included. The advantages of the velocity-potential formulation over the acceleration-potential formulation are discussed. The boundary-element algorithm used for the computational implementation is briefly outlined. Validation of the formulation is presented for airplane wings and helicopter rotors in hover. The test cases fall into two categories. prescribed-wake and free-wake analyses. The validation of the prescribed-wake analysis is presented for compressible flows, subsonic for helicopter rotors, transonic for airplanes. The numerical validation of the free-wake analysis of helicopter rotors is presented for incompressible flows.

Folded plates revisited

Abstract: Extensions of a previous paper ‘Modeling of a folded plate’ are considered. These include folded plates clamped along one edge only, plates folded at an arbitrary angle and structure with corners.

Vibration of cracked beams

Abstract: A surface crack on a beam section introduces a local flexibility to the structural member. This alters the vibration response of the system, shifting its frequency and increasing its mobility. Strain energy concentration arguments lead to the development of a compliance matrix for the behavior of the beam in the vicinity of the crack. This matrix is used to develop the transfer matrix for the cracked beam element. The transfer matrix developed, was used to evaluate the dynamic characteristics of simple cracked beams experiencing free vibrations. Reasonant frequencies and vibration modes are considerably affected by the existence of moderate cracks. Results for simple beams are presented for different boundary conditions.

Application of the symbolic mathematics system to the finite element program

Abstract: The stiffness matrix of a 4-nodes insoparametric element for the two dimensional finite element analysis is evaluated using the symbolic mathematics systems, called REDUCE The results are compared with those obtained by numerical integration It is shown that there exist some differences between the numerical values of the stiffness matrix by symbolic and numerical integrations The differences become smaller as the integration points of the numerical integration increases, which means that the results obtained by symbolic integration give accurate values The stiffness matrix in elastic-plastic state is also evaluated using symbolic mathematics, and it is shown that the results agree with those obtained by numerical integration It is pointed out that there is a possibility to decrease the CPU time by using the symbolic mathematics system

Dynamic analysis of polygonal Mindlin plates on two-parameter foundations using classical plate theory and an advanced BEM

Abstract: Forced vibrations of moderately thick plates on two-parameter, Pasternak-type foundations are considered. Influence of plate shear and rotatory inertia are taken into account according to Mindlin. Excitations are of the force as well as of the support motion type. Formulation is in the frequency domain. An analogy to thin plates without foundations is given. This analogy to classical plate theory is complete in the case of polygonal plan-forms and hinged support conditions. In that case the higher order Mindlin-problem is reduced to two (second order) Helmholtz-Klein- Gordon boundary value problems. An advanced BEM using Green's functions of rectangular domains is applied to the latter, thereby satisfying boundary conditions exactly as far as possible. This problem oriented strategy provides the frequency response functions for the deflection of the undamped Mindlin plate with high numerical accuracy. Structural damping is built in subsequently, and Fast Fourier Transform is applied for calculation of the transient response.

Space-time discontinuous finite element approximations for multi-dimensional nonlinear hyperbolic systems

Abstract: The discontinuous Galerkin method is applied on both spatial and temporal dimensions The new approach naturally extends, through flux splitting with an alternating sweep in the forward and backward space directions, the basic spirit of the Lesaint's scalar algorithm for multi-dimensional nonlinear hyperbolic systems The method is used to solve numerically a number of representative quasi-linear hyperbolic equations which involve propagation of a sharp front through the solution domain A posteriori analysis indicates that the proposed quasi-explicit scheme is unconditionally stable and for smooth solutions its rate of convergence is optimal Computationally the method appears to offer a number of advantages over other schemes including quite an accurate description of the structure of the shock front Several areas for future research are described

On the mechanical inconsistency of symmetrization of unsymmetric coupling matrices for BEFEM discretizations of solids

Abstract: For several technical problems discretization of the boundary of a part of a solid by the BEM and of the remaining part of the solid by the FEM is useful in order to exploit the complementary advantages of the BEM and of the FEM optimally. A characteristic feature of the employed “local FEM approach” for the coupling of such discretizations are “coupling matrices” representing load-stiffness matrices which are associated with displacement-dependent node forces acting on the interface of the two parts of the solid. Because of the nature of the employed BEM these matrices are unsymmetric. It is proved theoretically that symmetrization of such coupling matrices is mechanically inconsistent. It is also demonstrated that this symmetrization may lead to a significant deterioration of the quality of numerical results.

Application of the symbolic mathematics system to the finite element program: evaluation of the stiffness matrix of a 4-nodes isoparametric element

Abstract: The stiffness matrix of a 4-nodes isoparametric element for FEM analysis is evaluated by using the symbolic mathematics system. The results are compared with those by the numerical integration. It is shown that the stiffness matrix by the symbolic manipulation gives accurate results, though the results by the numerical integration have some numerical error.

Boundary integral equation formulations for free-surface flow problems in two and three dimensions

Abstract: To compute the transient solution of free-surface flow problems in two and three dimensions boundary integral equation formulations are considered. Consistent lower and higher order approximations based on small curvature expansions are compared and applied to a time-dependent, linear free-surface wave problem.

Physically based sensitivity derivatives for structural analysis programs

Abstract: Most formulations for sensitivity derivatives of structural response require detail-level computations (e.g. element-level calculations for finite-element systems). This requirement is a major difficulty in this age of “black box” commercial structural-analysis software which typically does not provide the user with access to that level of detail. A recent formulation of sensitivity analysis by Mroz and Dems provides a solution to this difficulty. The sensitivity calculations only require the solution of the original problem with new imposed loading consisting of initial stresses and initial strains. Therefore, they require only that the structural analysis program allow these type of loading. The present paper demonstrates the application of this approach to one and two-dimensional problems, using a finite-element program and a nonlinear shell analysis program.

Close interaction of coplanar circular cracks under shear loading

Abstract: A new method is proposed for the stress analysis of an elastic space weakened by several arbitrarily located coplanar circular cracks subjected to an arbitrary shear loading. The method is based on a new type of integral equation and has definite advantages over the existing methods: equations are non-singular, the iteration procedure is rapidly convergent even for very close interactions. The method allows us to obtain a practically exact numerical solution to the problem of very close interactions. An accurate analytical solution is obtained for the case of two cracks, which are separated by a half of their radius or more. The stress intensity factors and the crack energy increase due to the interaction are computed for various distances between the cracks.

Thermomechanical behaviour of composites

Abstract: This paper derives the governing equations for the thermomechanical behaviour of composites. When the basic equations for the thermoelastic behaviour of solids were first derived in the nineteenth century several approximations were made. The effect of these assumptions are discussed and illustrated by the results of a simple laboratory test. The implications of this work on the analysis of impact damaged laminates are then discussed.

Nonlinear natural vibration analysis of beams by selective coefficient increment

Abstract: The amplitude-frequency relation (backbone curve) of an elastic body in large amplitude natural vibration at resonance (including internal resonance) is of interest. The backbone curve is constructed by amplitude increment (decrement). Since the amplitude is given as a combination of harmonic components, the increment is performed actually on the coefficients of the harmonic components. An assumed increment is first applied to the first harmonic coefficient (the active harmonic) and a new equilibrium state is found by the Newtonian algorithm. The frequency increment is computed, all the harmonic coefficients are adjusted and the amplitude is evaluated. During the process, it is possible that one of the originally small harmonic coefficients suddenly increases in magnitude. The harmonic coefficient is selected to be the new active harmonic in the next step to find a new equilibrium state. The internal resonance of order 5 combining the first and the third linear modes of a clamped-clamped beam is predicted without difficulty. Therefore, automatic computation of combined resonance is straight forward. Matrix notation is used when possible to given concise presentation. The method is readily applicable in finite element sense.

Plastic deformation of inhomogeneous materials with elliptic inclusions

Abstract: A rigid-plastic finite element method which is based on the upper bound theory in plasticity is applied to the study of deformation behavior of inhomogeneous materials with inclusions. The penalty method and the Newton-Raphson's repeated calculation are adopted to minimize the functional and to obtain the solution. Characteristics of the deformation behavior of inhomogeneous material with inclusions are discussed based on the calculated results.

On finite axi-symmetrical deformations of thin elastic shells of revolution

Abstract: We derive a generalized version of the known system of two simultaneous second order differential equations for the problem of axi-symmetric torsionless deformations of elastic shells of revolution, for finite deformations and including transverse shear deformations and membrane drilling moments. Our generalization, which involves the introduction of a semicomplementary energy density, comes out in a particularly simple and compact form. We furthermore consider the effect of transverse normal stress deformations and discover the possibility of reducing this problem to a system of three simultaneous second order equations, with the supplementary third equation harmoniously adding itself to the two equations without consideration of transverse normal stress deformation effects.

Simulation of Rolling Processes by the Boundary Element Method

Abstract: Rolling processes play a very important role in everyday manufacturing. It is demonstrated here that the boundary element method (BEM) can be used to analyze, effectively and accurately, this class of problems involving both material and geometric nonlinearities, as well as contact boundary conditions. The BEM formulation is capable of using any of a class of combined creep-plasticity constitutive models with state variables for the description of material behavior. The specific problem considered is plane strain slab rolling using the constitutive model originally proposed by Hart. The numerical results obtained from the BEM analysis provide a lot of insights into the process and can become a useful tool in designing these rolling operations.

Approximation of general shell problems by flat plate elements

Abstract: The approximation of general shell problems by flat plate elements is very popular among engineering people. These methods have as common features a nonconforming approximation of the geometry of the considered shell using facet elements and a pseudo-conforming approximation of the components of the displacement, i.e., an approximation using conforming plate elements over every flat element. In this work, we analyze the compatibility conditions which have to be satisfied by the degrees of freedom at every node of the triangulation. Next, we obtain several interesting results valid for general shells and we prove the “pseudo-convergence” of the method for a class of shallow shells. Then, this careful study allows to introduce a perturbation of the bending terms upon each facet; the corresponding new method is convergent for arbitrary thin shells.

A stability analysis of the uniaxial viscoplasticity theory based on overstress

Abstract: In the development of a viscoplasticity theory without a yield surface and without a loading and unloading condition, the properties of a uniaxial constitutive model consisting of two coupled nonlinear differential equations are examined. The critical points of the system of differential equations are evaluated for monotonic loading, creep and relaxation conditions and are shown to be stable. The conditions necessary for the elimination of stable but oscillatory solutions are given. Also given are the asymptotic solutions valid near the critical points. The analytical predictions are confirmed by numerical results.

Boundary element method analysis of bending of inelastic plates with general boundary conditions

Abstract: This paper presents an accurate boundary element method (BEM) formulation for the bending of inelastic Kirchhoff plates subjected to general boundary conditions. This approach is an extension of earlier work by the authors of this paper and other co-workers on elastic plate deformation where they had proposed a three-equation BEM scheme. Numerical results presented here include plates with cutouts and free edges. A rate type constitutive model is used here to describe nonelastic deformation behavior of the plate material.

Thermodynamic constraints on stress rate formulations in constitutive models

Abstract: The principles of reversible thermodynamics can be used to place constraints on the form of the constitutive law in thermoelastic problems. The constraints indicate that the use of stress-rates can lead to complications in the form of the constitutive law that are removed when rates are not employed. In fact, in a large rotation small strain thermoelastic problem the stress can be written directly in terms of the symmetric part of the deformation gradient, and the temperature. The rotation at each point can be removed by the solution of an eigenvalue problem. For the large rotation large strain inelastic problem with varying temperatures, the work presented in this paper can be further developed for determining thermodynamically consistent constitutive models.