By Roland Glowinski, Patrick Le Tallec

A necessity for a deeper figuring out of the convergence houses of augmented Lagrangian algorithms and in their courting to operator-splitting tools corresponding to alternating-methods path and the advance of extra effective algorithms brought on the authors to write down this ebook. the amount is orientated to purposes in continuum mechanics. This quantity bargains with the numerical simulation of the habit of constant media by way of augmented Lagrangian and operator-splitting equipment (coupled to finite-element approximations). It starts with an outline of the mechanical and mathematical frameworks of the thought of functions in addition to a common research of the fundamental numerical equipment also used to check them. those principles are then utilized to precise periods of mechanical difficulties.

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A necessity for a deeper figuring out of the convergence homes of augmented Lagrangian algorithms and in their courting to operator-splitting tools equivalent to alternating-methods path and the improvement of extra effective algorithms caused the authors to write down this e-book. the quantity is orientated to purposes in continuum mechanics.

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2. Weak formulations. 2). This leads to a FIG. 2. 4). 40 CHAPTER 2 problem of the form which is completely specified as soon as the set K of kinematically admissible displacement fields and the space V of test functions (virtual displacements) are defined. 4) and proceeding by analogy with the linear case, we may choose The superscript s is such that the integral makes sense for any u in K and any v in V. 7) for K and V, corresponds to the classical weak formulation of equilibrium problems in compressible finite elasticity.

7. <+oo, be given. Let Fj be a measurable subset of F whose interior in T is not empty. 3. Basic tools for existence theory. Existence theory is important from the numerical point of view because it helps in the choice of consistent approximation techniques and numerical methods for the solution of the underlying problem. In this book, existence results will be based mainly on three fundamental theorems. 8. ) Let V be a Hilbert space, a ( - , - ) a continuous bilinear form on Vx V, and L ( - ) a continuous linear form on V.

The existence of solutions follows, then, from a direct application of the generalized Weierstrass theorem on W M (fl). To check the uniqueness of the solution when the internal dissipation potential is strictly convex, consider a solution v of the above problem, an arbitrary element w of Xj different from v, and compute J((w+v)/2). 1. 1 is still applicable to materials associated with potentials that are nondifferentiable and not strictly convex such as where Dt denotes the eigenvalues of D. This corresponds to Tresca-type viscoplasticity (see Chap.