Computational Geomechanics. Manuel Pastor

Computational Geomechanics - Manuel Pastor


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StartBinomialOrMatrix normal upper Delta ModifyingAbove Above bold u overbar With two-dots Subscript n Choose normal upper Delta ModifyingAbove Above bold p Superscript w Baseline overbar With ampersand c period dotab semicolon Subscript n EndBinomialOrMatrix Superscript l plus 1 Baseline equals StartBinomialOrMatrix normal upper Delta ModifyingAbove Above bold u overbar With two-dots Subscript n Choose normal upper Delta ModifyingAbove Above bold p Superscript w Baseline overbar With ampersand c period dotab semicolon Subscript n EndBinomialOrMatrix Superscript l Baseline plus StartBinomialOrMatrix delta normal upper Delta ModifyingAbove Above bold u overbar With two-dots Subscript n Choose delta normal upper Delta ModifyingAbove Above bold p Superscript w Baseline overbar With ampersand c period dotab semicolon Subscript n EndBinomialOrMatrix Superscript l plus 1

      The Jacobian matrix can be written as

      where

bold upper K Subscript normal upper T Baseline equals integral Underscript normal upper Omega Endscripts bold upper B Superscript normal upper T Baseline bold upper D Subscript normal upper T Baseline bold upper B d upper Omega plus ModifyingBelow integral Underscript normal upper Omega Endscripts left-parenthesis bold upper A upper B right-parenthesis Superscript normal upper T Baseline bold sigma prime prime bold upper A upper B d upper Omega With bar

      which are the well‐known expressions for tangent stiffness matrix. The underlined term corresponds to the “initial stress” matrix evaluated in the current configuration as a result of stress rotation defined in (2.5).

      Two points should be made here:

      1 that in the linear case, a single “iteration” solves the problem exactly

      2 that the matrix can be made symmetric by a simple scalar multiplication of the second row (provided KT is itself symmetric).

      In practice, it is found that the use of various approximations of the matrix J is advantageous such as, for instance, the use of “secant” updates (see, for instance, Crisfield (1979), Matthies, and Strang (1979) and Zienkiewicz et al (2013).

      A particularly economical computation form is given by choosing β2 = 0 and representing matrix M in a diagonal form. This explicit procedure was first used by Leung (1984) and Zienkiewicz et al. (1980a). It is, however, only conditionally stable and is efficient only for phenomena of short duration.

      The process of the time‐domain solution of (3.44) can be amended to that of successive separate solutions of the time equations for variables upper Delta ModifyingAbove Above bold u overbar With two-dots Subscript n and upper Delta ModifyingAbove Above bold p Superscript w Baseline overbar With ampersand c period dotab semicolon Subscript n, respectively, using an approximation for the remaining variable. Such staggered procedures, if stable, can be extremely economical as shown by Park and Felippa (1983) but the particular system of equations presented here needs stabilization. This was first achieved by Park (1983) and, later, a more effective form was introduced by Zienkiewicz et al. (1988).

      Special cases of solution are incorporated in the general solution scheme presented here without any modification and indeed without loss of computational efficiency.

      Thus, for static or quasi‐static, problems, it is merely necessary to put M = 0 and immediately the transient consolidation equation is available. Here time is still real and we have omitted only the inertia effects (although with implicit schemes, this a priori assumption is not necessary and inertia effects will simply appear as negligible without any substantial increase of computation). In pure statics, the time variable is still retained but is then purely an artificial variable allowing load incrementation.

Schematic illustration of elements used for coupled analysis, displacement (u) and pressure (p) formulation.
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