Variational principles of nonlinear fracture mechanics

Summary A variational principle of total energy is formulated for finite strain statics of a hyperelastic body whose initial configuration contains a gap. From this principle statical equations and boundary conditions for the gapped body are derived. The equilibrium condition at a gap tip is associated with the well-knownJ-integrals. By including the reaction of inertia and the flux of kinetic energy the principle of total energy is transformed to the variational inequality of evolution for dynamics of a hyperelastic body with a propagating crack. A closed system of dynamical equations, boundary conditions and additional conditions on the unknown contact crack surfaces and crack tip is obtained. As example the antiplane shear of an infinite gapped body is considered.     

A least squares finite element formulation for elastodynamic problems

A residual finite element formulation is developed in this paper to solve elastodynamic problems in which body wave potentials are primary unknowns. The formulation is based on minimizing the square of the residuals of governing equations as well as all boundary conditions. Since the boundary conditions in terms of wave potentials are neither Dirichlet nor Neumann type it is difficult to construct a functional to satisfy all governing equations and boundary conditions following the variational principle designed for conventional finite element formulation. That is why the least squares technique is sought. All boundary conditions are included in the functional expression so that the satisfaction of any boundary condition does not become a requirement of the trial functions, but they should satisfy some continuity conditions across the interelement boundary to guarantee proper convergence. In this paper it is demonstrated that the technique works well for elastodynamic problems; however, it is equally applicable to any other field problem.     

Equivalent histories,states and a variational problem for a rigid linear heat conductor with memory

The model of a rigid linear heat conductor which exhibits a constitutive equation with memory for the heat flux can be characterized by processes and states. The equivalence between histories is introduced in order to consider minimal states. The inversion of the constitutive equation, in which the heat flux appears as a linear functional of the history of the temperature gradient, allows to consider states expressed in terms of the heat flux vector instead of the temperature gradient. A variational formulation of an evolution problem with mixed initial boundary conditions is also given.     

Variational principles for nonstationary heat conduction problems

Integral variational principles are proposed for nonstationary heat conduction problems. These lead to integration of the wave equation of heat conduction or the Fourier heat conduction equation with the initial and boundary conditions.     

On the numerical implementation of variational arbitrary Lagrangian–Eulerian (VALE) formulations

This paper is concerned with the implementation of variational arbitrary Lagrangian–Eulerian formulations, also known as variational r‐adaption methods. These methods seek to minimize the energy function with respect to the finite‐element mesh over the reference configuration of the body. We propose a solution strategy based on a viscous regularization of the configurational forces. This procedure eliminates the ill‐posedness of the problem without changing its solutions, i.e. the minimizers of the regularized problems are also minimizers of the original functional. We also develop strategies for optimizing the triangulation, or mesh connectivity, and for allowing nodes to migrate in and out of the boundary of the domain. Selected numerical examples demonstrate the robustness of the solution procedures and their ability to produce highly anisotropic mesh refinement in regions of high energy density. Copyright © 2006 John Wiley & Sons, Ltd.     

Hyperbolic paraboloid shell analysis via mixed finite element formulation

An isoparametric rectangular mixed finite element is developed for the analysis of hypars. The theory of shallow thin hyperbolic paraboloid shells is based on Kirchhoff–Love's hypothesis and a new functional is obtained using the Gâteaux differential. This functional is written in operator form and is shown to be a potential. Proper dynamic and geometric boundary conditions are obtained. Applying variational methods to this functional, the HYP9 finite element matrix is obtained in an explicit form. Since only first-order derivatives occur in the functional, linear shape functions are used and a C⁰ conforming shell element is presented. Variation of the thickness is also included into the formulation without spoiling the simplicity. The formulation is applicable to any boundary and loading condition. The HYP9 element has four nodes with nine Degrees Of Freedom (DOF) per node—three displacements, three inplane forces and two bending, one torsional moment (4 × 9). The performance of this simple, and elegant shell element, is verified by applying it to some test problems existing in the literature. Since the element matrix is obtained explicitly, there is an important save of computer time.     

On a certain class of incorrectly stated problems in analytical theory of thermal conductivity and in the theory of indirect measurements

The variational problem of stationary thermal conductivity in an inhomogeneous solid is formulated. It is assumed that the boundary conditions on the boundary of the body are unknown. In order to obtain a unique and stable solution one requires measurement of the temperature at one point and correct selection of the regularization parameter.     

Variational principles for linear coupled dynamic theory of thermoviscoelasticity

Variational principles for linear coupled dynamic theory of thermoviscoelasticity are constructed using variational theory of potential operators. The functional derived herein gives, when varied, all the governing equations, including the boundary and initial conditions, as the Euler equations. The procedure shown herein does not require, in contrast to Gurtin's method, the transformation of field equations into an equivalent set of integro-differential equations, and includes the initial conditions of the problem explicitly. Gurtin's variational principle for dynamic theory of thermoviscoelasticity is also derived and compared with the present one. Variational principles for elastodynamics, visco-elasticity, etc. are derived as special cases of the variational principle derived herein.     

Complementary variational principles for steady heat conduction with mixed boundary conditions

Summary New stationary, maximum and minimum principles associated with the boundary value problem of steady heat conduction with general boundary conditions are derived in a unified manner from the theory of complementary variational principles. One of the results contains the Brand-Lahey [3] stationary principle as a special case.     

A shape optimization approach for simulating contact of elastic membranes with rigid obstacles

The obstacle problem consists in computing equilibrium shapes of elastic membranes in contact with rigid obstacles. In addition to the displacement u of the membrane, the interface Γ on the membrane demarcating the region in contact with the obstacle is also an unknown and plays the role of a free boundary. Numerical methods that simulate obstacle problems as variational inequalities share the unifying feature of first computing membrane displacements and then deducing the location of the free boundary a posteriori. We present a shape optimization-based approach here that inverts this paradigm by considering the free boundary to be the primary unknown and compute it as the minimizer of a certain shape functional using a gradient descent algorithm. In a nutshell, we compute Γ then u, and not u then Γ. Our approach proffers clear algorithmic advantages. Unilateral contact constraints on displacements, which render traditional approaches into expensive quadratic programs, appear only as Dirichlet boundary conditions along the free boundary. Displacements of the membrane need to be approximated only over the noncoincidence set, thereby rendering smaller discrete problems to be resolved. The issue of suboptimal convergence of finite element solutions stemming from the reduced regularity of displacements across the free boundary is naturally circumvented. Most importantly perhaps, our numerical experiments reveal that the free boundary can be approximated to within distances that are two orders of magnitude smaller than the mesh size used for spatial discretization. The success of the proposed algorithm relies on a confluence of factors- choosing a suitable shape functional, representing free boundary iterates with smooth implicit functions, an ansatz for the velocity of the free boundary that helps realize a gradient descent scheme and triangulating evolving domains with universal meshes. We discuss these aspects in detail and present numerous examples examining the performance of the algorithm.     

On the existence of optimal shapes in contact problems — Perfectly plastic bodies

The optimal shape design of a two-dimensional elastic perfectly plastic body (a punch) on a rigid frictionless foundation is analysed. The problem is to find the boundary part of the body where the unilateral boundary conditions are assumed in such a way that certain energy integral of the system in the equilibrium will be minimized. It is assumed that the material of the body is elastic perfectly plastic, obeying the Henckys law. The variational formulation in terms of stresses is utilized. The existence of optimal shapes is proved.     

Variational principles for steady heat conduction with mixed boundary conditions

Summary For the boundary value problem of steady heat conduction with general boundary conditions a variational problem is formulated by adding a simple surface integral to Butler's volume integral.This research was supported by the National Science Foundation, Grant GK-108.     

A lower bound for the two-dimensional Helmholtz equation

A variational expression (Temple's formula), requiring C₁ continuity, is used to transform the two-dimensional Helmholtz equation into a matrix equation. Finite element techniques are used and homogeneous Neumann or Dirichlet boundary conditions are assumed. It is demonstrated that under certain conditions lower bounds to the eigenvalues are obtained.     

DSC‐Ritz method for the free vibration analysis of Mindlin plates

This paper introduces a novel method for the free vibration analysis of Mindlin plates. The proposed method takes the advantage of both the local bases of the discrete singular convolution (DSC) algorithm and the pb‐2 Ritz boundary functions to arrive at a new approach, called DSC‐Ritz method. Two basis functions are constructed by using DSC delta sequence kernels of the positive type. The energy functional of the Mindlin plate is represented by the newly constructed basis functions and is minimized under the Ritz variational principle. Extensive numerical experiments are considered by different combinations of boundary conditions of Mindlin plates of rectangular and triangular shapes. The performance of the proposed method is carefully validated by convergence analysis. The frequency parameters agree very well with those in the literature. Numerical experiments indicate that the proposed DSC‐Ritz method is a very promising new method for vibration analysis of Mindlin plates. Copyright © 2004 John Wiley & Sons, Ltd.     

Macro-elements in the mixed boundary value problems

 The symmetric Galerkin boundary element method (SGBEM), applied to elastostatic problems, is employed in defining a model with BE macro-elements. The model is governed by symmetric operators and it is characterized by a small number of independent variables upon the interface between the macro-elements. The kinematical and mechanical transmission conditions across the interface are imposed in global form, whereas the response of the boundary discretized elastic problem is provided in terms of forces on the interface boundary sides and of displacements at the interface nodes. A variational formulation is presented in which the boundary transmission conditions are derived by Polizzotto's boundary min-max variational principle. Simple numerical applications are shown. Received 8 December 99     

On the symmetry and coaxiality of pertinent stretch and stress tensors during non-proportional loading at finite plastic strain

Summary In this paper we develop agradient theory of internal variables using a variational principle in conjunction with the dissipation inequality. The basic findings are (i), that the internal variables are, non-local in that they obey field equations instead of evolution equations and (ii) they are subject to boundary conditions that are dictated by the applied tractions and displacements as well as the physical structure of the material domain. As a consequence, spatiallyinhomogeneous strain fields exist in the presence ofuniform boundary tractions and/or displacements. This phenomenon is illustrated in the simple case of one dimension.     

Inverse mass matrix for isogeometric explicit transient analysis via the method of localized Lagrange multipliers

A variational framework is employed to generate inverse mass matrices for isogeometric analysis (IGA). As different dual bases impact not only accuracy but also computational overhead, several dual bases are extensively investigated. Specifically, locally discontinuous biorthogonal basis functions are evaluated in detail for B-splines of high continuity and Bézier elements with a standard C⁰ continuous finite element structure. The boundary conditions are enforced by the method of localized Lagrangian multipliers after generating the inverse mass matrix for completely free body. Thus, unlike inverse mass matrix methods without employing the method of Lagrange multipliers, no modifications in the reciprocal basis functions are needed to account for the boundary conditions. Hence, the present method does not require internal modifications of existing IGA software structures. Numerical examples show that globally continuous dual basis functions yield better accuracy than locally discontinuous biorthogonal functions, but with much higher computational overhead. Locally discontinuous dual basis functions are found to be an economical alternative to lumped mass matrices when combined with mass parameterization. The resulting inverse mass matrices are tested in several vibration problems and applied to explicit transient analysis of structures.     

Vibration analysis of stepped rectangular plates using the extended Kantorovich method

In this study, vibration behaviors of stepped plates are investigated based on the variational principle of minimum total energy and the extended Kantorovich method. The out-of-plane displacement is represented by a separable function of parameters x and y. A set of governing equations, boundary conditions, and continuity conditions in the form of ordinary differential equations are derived from the energy condition. The natural frequency and out-of-plane displacement function can be determined numerically from the derived equations and conditions. Solutions from the proposed approach are in good agreement with results from past studies and those of the finite element method.     

A mixed variational principle for finite element analysis

A mixed variational principle is developed and utilized in a finite element formulation. The procedure is mixed in the sense that it is based upon a combination of modified potential and complementary energy principles. Compatibility and equilibrium are satisfied throughout the domain a priori, leaving only the boundary conditions to be satisfied by the variational principle. This leads to a finite element model capable of relaxing troublesome interelement continuity requirements. The nodal concept is also abandoned and, instead, generalized parameters serve as the degrees-of-freedom. This allows for easier construction of higher order elements with the displacements and stresses treated in the same manner. To illustrate these concepts, plane stress and plate bending analyses are presented.