Computational Geomechanics. Manuel Pastor

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Computational Geomechanics - Manuel Pastor

Computational Geomechanics - Manuel Pastor

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target="_blank" rel="nofollow" href="#fb3_img_img_53e33fd9-bd27-5869-b0f7-01d4cf535b65.png" alt="StartLayout 1st Row ModifyingAbove Above bold p Superscript w Baseline overbar With ampersand c period dotab semicolon Subscript n plus 1 Baseline equals ModifyingAbove Above bold p Superscript w Baseline overbar With ampersand c period dotab semicolon Subscript n Baseline plus normal upper Delta ModifyingAbove Above bold p Superscript w Baseline overbar With ampersand c period dotab semicolon Subscript n Baseline 2nd Row bold p Superscript w Baseline overbar Subscript n plus 1 Baseline equals bold p Superscript w Baseline overbar Subscript n Baseline plus ModifyingAbove Above bold p Superscript w Baseline overbar With ampersand c period dotab semicolon Subscript n Baseline normal upper Delta t plus beta overbar Subscript 1 Baseline normal upper Delta ModifyingAbove Above bold p Superscript w Baseline overbar With ampersand c period dotab semicolon Subscript n Baseline normal upper Delta t EndLayout"/>

beta 2 greater-than-or-equal-to beta 1 greater-than-or-equal-to one half and beta overbar Subscript 1 Baseline greater-than-or-equal-to one half

      The optimal choice of these values is a matter of computational convenience, the discussion of which can be found in literature. In practice, if the higher order accurate “trapezoidal” scheme is chosen with β2 = β1 = 1/2 and beta overbar Subscript 1 = 1/2, numerical oscillation may occur if no physical damping is present. Usually, some algorithmic (numerical) damping is introduced by using such values as

beta 2 equals 0.605 beta 1 equals 0.6 and beta overbar Subscript 1 Baseline equals 0.6

      or

beta 2 equals 0.515 beta 1 equals 0.51 and beta overbar Subscript 1 Baseline equals 0.51 period

      Dewoolkar (1996), using the computer program SWANDYNE II in the modelling of a free‐standing retaining wall, reported that the first set of parameters led to excessive algorithmic damping as compared to the physical centrifuge results. Therefore, the second set was used and gave very good comparisons. However, in cases involving soil, the physical damping (viscous or hysteretic) is much more significant than the algorithmic damping introduced by the time‐stepping parameters and the use of either sets of parameters leads to similar results.

      This set can be written as

      (3.44a)bold upper Psi Subscript n plus 1 Superscript left-parenthesis 1 right-parenthesis Baseline equals bold upper M Subscript n plus 1 Baseline normal upper Delta ModifyingAbove Above bold u overbar With two-dots Subscript n Baseline plus bold upper P left-parenthesis bold u overbar Subscript n plus 1 Baseline right-parenthesis minus bold upper Q Subscript n plus 1 Baseline beta overbar Subscript 1 Baseline normal upper Delta t normal upper Delta ModifyingAbove Above bold p overbar With ampersand c period dotab semicolon Subscript n Superscript w Baseline minus bold upper F Subscript n plus 1 Superscript left-parenthesis 1 right-parenthesis Baseline equals 0

      (3.44b)bold upper Psi Subscript n plus 1 Superscript left-parenthesis 2 right-parenthesis Baseline equals bold upper Q overTilde Subscript n plus 1 Superscript normal upper T Baseline beta 1 normal upper Delta t normal upper Delta ModifyingAbove Above bold u overbar With two-dots Subscript n Baseline plus bold upper H Subscript n plus 1 Baseline beta overbar Subscript 1 Baseline normal upper Delta t normal upper Delta ModifyingAbove Above bold p overbar With ampersand c period dotab semicolon Subscript n Superscript w Baseline plus bold upper S Subscript n plus 1 Baseline normal upper Delta ModifyingAbove Above bold p overbar With ampersand c period dotab semicolon Subscript n Superscript w Baseline minus bold upper F Subscript n plus 1 Superscript left-parenthesis 2 right-parenthesis Baseline equals 0

      where bold upper F Subscript n plus 1 Superscript left-parenthesis 1 right-parenthesis and bold upper F Subscript n plus 1 Superscript left-parenthesis 2 right-parenthesis can be evaluated explicitly from the information available at time tn and

      (3.45)ModifyingAbove bold upper P With bar left-parenthesis bold u overbar Subscript n plus 1 Baseline right-parenthesis equals integral Underscript normal upper Omega Endscripts bold upper B Superscript normal upper T Baseline bold sigma double-prime Subscript n plus 1 Baseline d upper Omega equals integral Underscript normal upper Omega Endscripts bold upper B Superscript normal upper T Baseline normal upper Delta bold sigma Subscript n Superscript double-prime Baseline d upper Omega plus bold upper P left-parenthesis bold u overbar Subscript n Baseline right-parenthesis

      In this, normal upper Delta bold sigma double-prime Subscript n must be evaluated by integrating (3.27) as the solution proceeds. The values of bold u overbar n+1 and bold p Superscript w Baseline overbar n+1 at the time tn+1 are evaluated by Equation (3.43).

      The equation will generally need to be solved by a convergent, iterative process using some form of Newton–Raphson procedure typically written as

      (3.46a)Скачать книгу

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