Angelo Carini

A variational approach to boundary element elastodynamic analysis and extension to multidomain problems

Abstract Time-dependent discontinuities of both tractions and displacements are considered on the boundary of a homogeneous linear-elastic solid conceived as embedded in the elastic space. The analysis of the dynamic response of the solid to external actions is formulated in the time domain by means of suitable integral representations of boundary displacements and of boundary tractions using time-dependent fundamental solutions. The integral operator generated is proven to be symmetric in both space and time with respect to a time-convolutive bilinear form. As a consequence, the history of boundary tractions and displacements along any assigned time interval 0 is shown to be characterized by a variational theorem. The variational formulation developed is generalized to zonewise homogeneous systems and to elastically constrained bodies. The linear algebraic equations generated by boundary element modelling in space and time turn out to exhibit symmetry of the coefficient matrix.

Fundamental solutions for linear viscoelastic continua

Abstract The full set of fundamental solutions for the linear viscoelastic problem is derived from the relevant elastic fundamental solutions using the well-known so-called “correspondence principle”. The fundamental solutions are given for the 3D-continuum, the 2D-plane strain and the 2D-plane stress problem using a wide range of linear viscoelastic constitutive law models. Finally, the possible fields of application of the determined fundamental solutions are given using symmetric BIE formulations in space and time.

Boundary integral equation analysis in linear viscoelasticity: variational and saddle point formulations

This paper presents a symmetric, double integration boundary integral equation (BIE) approach to linear viscoelastic analysis. The time evolution of the boundary unknowns is shown to be characterized by a variational theorem over an arbitrary time interval and by a min-max theorem from the time origin to infinite (minimum for displacements, maximum for tractions). Boundary element (BE) discretization in space and time leads to algebraic equations with symmetric coefficient matrix.

Some variational formulations for continuum nonlinear dynamics

Abstract A modification of a theory developed by Tonti (1984) for obtaining extremal formulations of a generic nonlinear problem is applied to derive several variational formulations for the nonlinear continuum dynamic problem with prescribed initial conditions. Such a problem does not admit “classical” variational statements, owing to its lack of symmetry with respect to “classical” bilinear forms. However, Tontis theory, with some developments introduced first by Carini, and then in this work, allows the systematic derivation of variational statements which may prove helpful both in understanding theoretical aspects and in devising numerical integration schemes.

Variational formulation of the boundary element method in transient heat conduction

Transient heat conduction, as a prototype of diffusion problems, is considered over homogeneous domains with mixed boundary conditions (on temperature and flux). Distributed time dependent discontinuities of flux and of temperature are taken as unknown sources over the Dirichlet (given temperature) and Neumann (given flux) boundaries, respectively. The boundary integral operator thus arising, is shown to be symmetric in space and time with respect to a time-convolutive bilinear form. A variational principle is then derived which characterizes the boundary solution in time. Discretizations in space and time lead to linear equations with a symmetric coefficient matrix; solution procedures are briefly discussed.

Implementation of a symmetric boundary element method in transient heat conduction with semi-analytical integrations

A time-convolutive variational hypersingular integral formulation of transient heat conduction over a 2-D homogeneous domain is considered. The adopted discretization leads to a linear equation system, whose coefficient matrix is symmetric, and is generated by double integrations in space and time. Assuming polynomial shape functions for the boundary unknowns, a set of compact formulae for the analytical time integrations is established. The spatial integrations are performed numerically using very efficient formulae just recently proposed. The competitiveness from the computational point of view of the symmetric boundary integral equation approach proposed herein is investigated on the basis of an original computer implementation. Copyright

Time integration errors and some new functionals for the dynamics of a free mass

Abstract We study the numerical integration of the Poisson second-order ordinary differential equation which describes, for instance, the dynamics of a free mass. Classical integration algorithms, when applied to such an equation, furnish solutions affected by a significant “drift” error, apparently not studied so far. In the first part of this work we define measures of such a drift. We then proceed to illustrate how to construct both classical and extended functionals for the equation of motion of a free mass with given initial conditions. These tools allow both the derivation of new variationally-based time integration algorithms for this problem, and, in some cases, the theoretical isolation of the source of the drift. While we prove that this particular error is unavoidable in any algorithmic solution of this problem, we also provide some new time integration algorithms, extensions at little added cost of classical methods, which permit to substantially improve numerical predictions.

Colonnetti's minimum principle extension to generally non-linear materials

Abstract In this paper two minimum principles are presented for the continuum problem with general non-linear materials (holonomic, non-holonomic, with hardening or softening, time-dependent etc. The proof of the first principle is based on the use of an elastic auxiliary problem associated to the original non-linear one and on the interpretation of the actual inelastic strains as unknown strains imposed on the elastic auxiliary solid. In a dual way the proof of the second principle is given through the imposition of suitable stresses on the elastic auxiliary solid. Classical principles of elasticity and incremental elastoplasticity are then derived from the new principles as particular cases. Three simple illustrative examples are given.

On the construction of extended problems and related functionals for general nonlinear equations

Abstract Starting from existing methods for the symmetrisation of general nonlinear, nonpotential operators (Tonti, Int. J. Engng. Sci. 22 (11–12) (1984) 1343–1371; Auchmuty, Nonlinear Anal. Theory Methods Appl. 12 (5) (1988) 531–564) this work discusses some alternative formulations and illustrates some significant implications of such methods, which should make them more suited to practical application. Further, a new class of the so-called “extended” functionals is proposed, much simpler to construct than the preceding ones. Even if the definition of the functionals requires the doubling of the unknown functions, the new unknowns have a precise physical meaning in the solution of the problem, which may help in the actual solution process. The application of the new method is illustrated by means of two examples in the field of continuum mechanics: the nonassociated elastic–plastic rate constitutive equations, and the nonlinear continuum dynamics equations with initial conditions.

Saddle-point principles and numerical integration methods for second-order hyperbolic equations

Abstract This work describes a family of functionals whose stationarity – often saddle-point condition – leads to well-known so-called “variational” formulations for structural dynamics (such as the weak Hamilton/Ritz formulation and the continuous/discontinuous Galerkin formulation) and, in turn, to methods for the numerical integration of the equations of motion. It is shown that all the time integration methods based on “variational” formulations do descend from such functionals. Moreover, starting from the described family of functionals it is possible to construct new families of time integration methods, which might exhibit computational advantages over the corresponding ones derived from “variational” formulations only.