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The finite difference method doesn’t postulate explicitly any specific shape of the unknown field. As we are concerned with partial differential equations, exact deriva­tives are replaced by an approximation based on neighbouring values of the un­known (still denoted as p): P,+1 — Pi-1 W/, 2L where the subscript i denotes the cell number and L […]

The basic idea of the finite element method is to divide the field to be analysed into sub-domains, the so-called finite elements, of simple shape: e. g. triangles, quadrilaterals with linear, parabolic or cubic sides for two-dimensional analysis. In each finite element, an analytically simple equation is postulated for the vari­able to be determined, i. […]

11.2.1 Introduction An approximated solution of most problems described by a set of partial differential equations may be obtained by numerical methods like the finite element method (FEM), the discrete element method (DEM), the finite difference method (FDM), the finite volume method (FVM), or the boundary element method (BEM). For the problems concerned here, the […]

In the preceding section, differential equations were given for three types of prob­lems. In order to solve these equations, we need to define boundary and initial con­ditions. Classical boundary conditions may be considered: imposed displacements or forces for solid mechanics problems and imposed fluid pressures, temperatures, concentrations or fluxes for diffusion and advection-diffusion problems. However, […]

Transport of pollutant or of heat in porous media is governed by a combination of advection and diffusion (Chapter 6, Section 6.3.1). The advection phenomenon is related to the transport (noted as a flow f ) of any substance by a fluid flow, described by the fluid’s velocity, f df/: fad. = Cf (11.11) The […]

Thermal conduction exchanges (Chapter 4) in solids and diffusion of contaminants (Chapter 6) are modelled by similar diffusion equations. The balance equation is written: дг/г + Q — S (11.6) where / represents a flux of fluid or heat, Q represents a sink term and S represents the storage of fluid or of heat. When […]

On the one hand, solid mechanics can be modelled on the following basis. The equilibrium equation is: дг <jjj + Pj = 0 (11.1) Where Pj is a member of P, the vector of volume forces, ji;- is a member of j, the Cauchy stress tensor, and д represents the spatial partial derivative operator: di […]

When trying to replicate in-situ behaviour by computational techniques, a number of different physical phenomena (Gens, 2001) need to be considered, including: • The non-linear solid mechanics and especially granular unbound or bound mate­rial mechanics: we consider the relations between displacements, strains, stresses and forces within solids. The material behaviour is described by a constitutive […]

Robert Charlier[25], Lyesse Laloui, Mihael Brencic, SigurSur Erlingsson, Klas Hansson and Pierre Hornych Abstract Different physical problems have been analysed in the preceding chapters: they relate to water transfer, to heat transfer, to pollutant transfer and to mechanical equilibrium. All these problems are governed by differential equations and boundary conditions but analytical solutions are, in […]

This Chapter presents in-situ and laboratory experimental techniques used to de­scribe mechanical behaviour of pavement materials (soils and aggregates) at dif­ferent saturation stages. Repeated triaxial load testing can be applied to obtain both stiffness characteristics and assessments of the ability of the material to with­stand accumulation of permanent deformation during cyclic loading. For unsatu­rated soils, […]