By Professor Roland Glowinski, Philippe G Ciarlet, Jacques Louis Lions, J L Lions
ISBN-10: 0080507948
ISBN-13: 9780080507941
ISBN-10: 0444512241
ISBN-13: 9780444512246
Hardbound. This publication dimension article is devoted to the numerical simulation of unsteady incompressible viscous circulation modelled via the Navier - Stokes equations, or by way of non-Newtonian editions of them. that allows you to accomplish that objective we've built a strategy in keeping with the next instruments:
(i) Time discretization by way of operator splitting schemes comparable to Peaceman - Rachford's, Douglas - Rachford's ,Marchuk - Yanenko's, Strang's symmetrized, and the so-called theta - scheme brought through the writer within the mid - eighties.
(ii) Projection equipment (in L2 or H1) for the remedy of the incompressibility div u = 0.
(iii) therapy of the advection by way of: both a headquartered scheme resulting in linear or nonlinear advection - diffusion difficulties solved via least squares / conjugate gradient algorithms, or to a linear wave - like equation compatible to finite aspect established answer equipment.
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Extra info for Handbook of Numerical Analysis. Numerical Methods for Fluids
Example text
A good example of what can go wrong is provided by a linear spring: although the displacement caused by forces abides by the principle of superposition, the stored elastic energy does not. Indeed, the latter is a quadratic function of the forces. 6 An interesting observation pertaining to the method of finite elements in Solid Mechanics can be made: to express the stresses in components on the same basis as the displacements leads to stress fields with artificial discontinuities. Vector and Affine Spaces Similarly, the multiplication of a scalar α by a linear function f is, by definition, the linear function αf given by: (αf )(v) := α f (v), ∀ v ∈ V.
From the physical point of view the cotangent space is a veritable bonanza: it contains all those physical entities which produce linearly a real number δW, called virtual work, when acting on virtual displacements. And what can these entities be but generalized forces acting on the configuration κ? There can be no other meaning of a generalized force. This being so, we have, by merely following the natural geometric constructions arising from the very definition of the configuration space Q, stumbled upon the most general possible notion of generalized force in Continuum Mechanics.
For the case U = V, contrast this last result with the expression for the action of the covector on the vector. [Remark: note, in passing, that not every linear operator can be expressed as the tensor product of two vectors. ] We have spoken about the conceptual novelty of the tensor product. No less important is its notational convenience. In the case of a vector, say v ∈ V, it is obviously advantageous to be able to express the master concept of a vector in terms of the subsidiary notion of its components in a particular basis by simply writing: v = v i fi .
Handbook of Numerical Analysis. Numerical Methods for Fluids by Professor Roland Glowinski, Philippe G Ciarlet, Jacques Louis Lions, J L Lions
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