From 7b47558ae4cf8f8df8457fffcbde0e654ff88754 Mon Sep 17 00:00:00 2001 From: mcbride Date: Tue, 21 Feb 2012 17:16:37 +0000 Subject: [PATCH] step:44 updated introduction, corrected tex, minor changes in cc git-svn-id: https://svn.dealii.org/trunk@25133 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-44/doc/intro.dox | 44 +++++++++++++++----------- deal.II/examples/step-44/step-44.cc | 7 ++-- 2 files changed, 28 insertions(+), 23 deletions(-) diff --git a/deal.II/examples/step-44/doc/intro.dox b/deal.II/examples/step-44/doc/intro.dox index 572474a108..329dc30b0a 100644 --- a/deal.II/examples/step-44/doc/intro.dox +++ b/deal.II/examples/step-44/doc/intro.dox @@ -1,3 +1,11 @@ +
+ +This program was contributed by Jean-Paul Pelteret and Andrew McBride. +
+This material is based upon work supported by the German Science Foundation (Deutsche +Forschungsgemeinschaft, DFG), grant STE 544/39-1, and the National Research Foundation of South Africa. +
+

Introduction

@@ -322,15 +330,15 @@ The three-field variational principle used here is given by where the external potential is defined by @f[ \Pi_{\textrm{ext}} - = - \int_\Omega \mathbf{b} \cdot \mathbf{u}~\textrm{d}v - - \int_{\partial \Omega_{\sigma}} \overline{\mathbf{t}} \cdot \mathbf{u}~\textrm{d}a \, . + = - \int_\Omega \mathbf{b}^\text{p} \cdot \mathbf{u}~\textrm{d}v + - \int_{\partial \Omega_{\sigma}} \mathbf{t}^\text{p} \cdot \mathbf{u}~\textrm{d}a \, . @f] The boundary of the current configuration $\partial \Omega$ is composed into two parts as $\partial \Omega = \partial \Omega_{\mathbf{u}} \cup \partial \Omega_{\sigma}$, where $\partial \Omega_{\mathbf{u}} \cap \partial \Omega_{\boldsymbol{\sigma}} = \emptyset$. -The prescribed Cauchy traction, denoted $\overline{\mathbf{t}}$, is applied to $ \partial \Omega_{\boldsymbol{\sigma}}$ while the motion is prescribed on the remaining portion of the boundary $\partial \Omega_{\mathbf{u}}$. -The body force per unit current volume is denoted $\mathbf{b}$. +The prescribed Cauchy traction, denoted $\mathbf{t}^\text{p}$, is applied to $ \partial \Omega_{\boldsymbol{\sigma}}$ while the motion is prescribed on the remaining portion of the boundary $\partial \Omega_{\mathbf{u}}$. +The body force per unit current volume is denoted $\mathbf{b}^\text{p}$. @@ -351,12 +359,12 @@ The stationarity of the potential follows as -\widetilde{p}\right] \right\}~\textrm{d}V \\ - &\quad - \int_{\Omega_0} \delta \mathbf{u} \cdot \mathbf{b}~\textrm{d}v - - \int_{\partial \Omega_{0,\boldsymbol{\sigma}}} \mathbf{u} \cdot \overline{\mathbf{t}}~\textrm{d}a + &\quad - \int_{\Omega_0} \delta \mathbf{u} \cdot \mathbf{B}^\text{p}~\textrm{d}V + - \int_{\partial \Omega_{0,\boldsymbol{\sigma}}} \delta \mathbf{u} \cdot \mathbf{T}^\text{p}~\textrm{d}A \\ &=0 \, , @f} -for all virtual displacements $\delta \mathbf{u} \in H^1(\Omega)$ subject to the constraint that $\mathbf{u} = \mathbf{0}$ on $\partial \Omega_{\mathbf{u}}$, and all virtual pressures $\delta p \in L^2(\Omega)$ and virtual dilatations $\delta \widetilde{J} \in L^2(\Omega)$. +for all virtual displacements $\delta \mathbf{u} \in H^1(\Omega)$ subject to the constraint that $\mathbf{u} = \mathbf{0}$ on $\partial \Omega_{\mathbf{u}}$, and all virtual pressures $\delta \widetilde{p} \in L^2(\Omega)$ and virtual dilatations $\delta \widetilde{J} \in L^2(\Omega)$. One should note that the definitions of the volumetric Cauchy stress and the subsequent tangent differs slightly from the general form given in the section on hyperelastic materials. This is because the pressure $\widetilde{p}$ is now a primary field. Note that although the variables are all expressed in terms of spatial quantities, the domain of integration is the reference configuration. @@ -364,14 +372,14 @@ This approach is called a total-Lagrangian formulation. The approach given in step-18 could be called updated Lagrangian. The Euler-Lagrange equations corresponding to the residual are: @f{align*} - &\textrm{div}\ \boldsymbol{\sigma} + \mathbf{b} = \mathbf{0} && \textrm{[equilibrium]} + &\textrm{div}\ \boldsymbol{\sigma} + \mathbf{b}^\text{p} = \mathbf{0} && \textrm{[equilibrium]} \\ &J(\mathbf{u}) = \widetilde{J} && \textrm{[dilatation]} \\ &\widetilde{p} = \dfrac{\textrm{d} \Psi_{\textrm{vol}}(\widetilde{J})}{\textrm{d} \widetilde{J}} && \textrm{[pressure]} \, . @f} The first equation is the equilibrium equation in the spatial setting. -The second is the constraint that $J(\mathbf{u}) = \widetilde{J}$, i.e., the incompressibility. +The second is the constraint that $J(\mathbf{u}) = \widetilde{J}$. The third is the definition of the pressure $\widetilde{p}$. We will use the iterative Newton-Raphson method to solve the nonlinear residual equation $R$. @@ -416,32 +424,32 @@ where @f{align*} D_{\varDelta \mathbf{u}} R( \mathbf{\Xi}; \delta \mathbf{\Xi}) &= - \int_\Omega \bigl[ \textrm{grad}\ \delta \mathbf{u} : + \int_{\Omega_0} \bigl[ \textrm{grad}\ \delta \mathbf{u} : \textrm{grad}\ \varDelta \mathbf{u} [\boldsymbol{\tau}_{\textrm{iso}} + \boldsymbol{\tau}_{\textrm{vol}}] + \textrm{grad}\ \delta \mathbf{u} :[ \underbrace{[\widetilde{p}J[\mathbf{I}\otimes\mathbf{I} - 2 \mathcal{I}]}_{\equiv J\mathfrak{c}_{\textrm{vol}}} + J\mathfrak{c}_{\textrm{iso}}] :\textrm{grad} \varDelta \mathbf{u} \bigr]~\textrm{d}V \, , \\ - &\quad + \int_\Omega \delta \widetilde{p} J \mathbf{I} : \textrm{grad}\ \varDelta \mathbf{u} ~\textrm{d}V + &\quad + \int_{\Omega_0} \delta \widetilde{p} J \mathbf{I} : \textrm{grad}\ \varDelta \mathbf{u} ~\textrm{d}V \\ D_{\varDelta \widetilde{p}} R( \mathbf{\Xi}; \delta \mathbf{\Xi}) &= - \int_\Omega \textrm{grad}\ \delta \mathbf{u} : J \mathbf{I} \varDelta \widetilde{p} ~\textrm{d}V - - \int_\Omega \delta \widetilde{J} \varDelta \widetilde{p} ~\textrm{d}V \, , + \int_{\Omega_0} \textrm{grad}\ \delta \mathbf{u} : J \mathbf{I} \varDelta \widetilde{p} ~\textrm{d}V + - \int_{\Omega_0} \delta \widetilde{J} \varDelta \widetilde{p} ~\textrm{d}V \, , \\ D_{\varDelta \widetilde{J}} R( \mathbf{\Xi}; \delta \mathbf{\Xi}) - &= -\int_\Omega \delta \widetilde{p} \varDelta \widetilde{J}~\textrm{d}V - + \int_\Omega \delta \widetilde{J} \dfrac{\textrm{d}^2 \Psi_{\textrm{vol}}(\widetilde{J})}{\textrm{d} \widetilde{J}\textrm{d}\widetilde{J}} \varDelta \widetilde{J} ~\textrm{d}V \, . + &= -\int_{\Omega_0} \delta \widetilde{p} \varDelta \widetilde{J}~\textrm{d}V + + \int_{\Omega_0} \delta \widetilde{J} \dfrac{\textrm{d}^2 \Psi_{\textrm{vol}}(\widetilde{J})}{\textrm{d} \widetilde{J}\textrm{d}\widetilde{J}} \varDelta \widetilde{J} ~\textrm{d}V \, . @f} Note that the following terms are termed the geometrical stress and the material contributions to the tangent matrix: @f{align*} -& \int_\Omega \textrm{grad}\ \delta \mathbf{u} : +& \int_{\Omega_0} \textrm{grad}\ \delta \mathbf{u} : \textrm{grad}\ \varDelta \mathbf{u} [\boldsymbol{\tau}_{\textrm{iso}} + \boldsymbol{\tau}_{\textrm{vol}}]~\textrm{d}V && \quad {[\textrm{Geometrical stress}]} \, , \\ -& \int_\Omega \textrm{grad} \delta \mathbf{u} : +& \int_{\Omega_0} \textrm{grad} \delta \mathbf{u} : [J\mathfrak{c}_{\textrm{vol}} + J\mathfrak{c}_{\textrm{iso}}] :\textrm{grad}\ \varDelta \mathbf{u} ~\textrm{d}V && \quad {[\textrm{Material}]} \, . @@ -464,7 +472,7 @@ and a classical displacement based method is recovered. For fully-incompressible materials $\nu = 0.5$ and the three-field formulation will still exhibit locking behaviour. This can be overcome by introducing an additional constraint into the free energy of the form -$\int_\Omega \Lambda [ \widetilde{J} - 1]~\textrm{d}V$. +$\int_{\Omega_0} \Lambda [ \widetilde{J} - 1]~\textrm{d}V$. Here $\Lambda$ is a Lagrange multiplier to enforce the isochoric constraint. For further details see Miehe (1994). diff --git a/deal.II/examples/step-44/step-44.cc b/deal.II/examples/step-44/step-44.cc index 560a4fd7a9..75f0879c77 100644 --- a/deal.II/examples/step-44/step-44.cc +++ b/deal.II/examples/step-44/step-44.cc @@ -215,8 +215,6 @@ namespace Step44 // The use of an effective preconditioner is critical to ensure // convergence when a large nonlinear motion occurs // in a Newton increment. -// ToDo: explain -// The default values are optimal for single-thread conditions this particular problem. struct LinearSolver { std::string type_lin; @@ -2230,9 +2228,8 @@ namespace Step44 // Calculate how well the dilatation $\widetilde{J}$ agrees with $J := // \textrm{det}\ \mathbf{F}$ from the $L^2$ error $ \bigl[ \int_{\Omega_0} {[ J -// - \widetilde{J}]}^{2}\textrm{d}V \bigr]^{1/2}$ which is then normalised by -// the current volume $\int_{\Omega_0} J ~\textrm{d}V = \int_\Omega -// ~\textrm{d}v$. We also return the ratio of the current volume of the +// - \widetilde{J}]}^{2}\textrm{d}V \bigr]^{1/2}$. +// We also return the ratio of the current volume of the // domain to the reference volume. This is of interest for incompressible // media where we want to check how well the isochoric constraint has been // enforced. -- 2.39.5