The new approach for a model problem in perfect plasticity

The approach to adaptive error control in finite element methods described above can easily be extended to nonlinear situations. We illustrate this for the Strang problem introduced above. The dual-mixed variational formulation of (1) reads

$$\sigma \tau \sigma \tau \sigma \geq \forall \tau \in \Pi$$ (12)

$$\sigma \varphi \varphi \forall \varphi \in \Omega$$ (13)

where

$$\Pi$$H^div = {$$\tau$$ $$\in$$ L²($$\Omega$$)² ,  div$$\tau$$ $$\in$$ L²($$\Omega$$),  |$$\tau$$|$$\le$$1  a.e. in  $$\Omega$$} .

Integrating in (12) by parts, we obtain the primal-mixed formulation

$$\sigma \tau \sigma \nabla \tau \sigma \geq \forall \tau \in \Pi \Omega$$ (14)

$$\sigma \nabla \varphi \varphi \forall \varphi \in$$ (15)

where  $\Pi$L²($\Omega$)²  has the obvious meaning. Although, the primal-mixed formulation is not properly posed, as it requires more regularity for the deformation,  u $\in$ H¹($\Omega$), than what can be expected in general, it has a feature which makes it particular attractive as starting point for the numerical approximation. The first relation in (13) can equivalently be written in the form

$$\sigma \nabla \Omega$$ (16)

with the function

$$\tau \Pi \tau \begin{cases} \quad \tau \,, &\text{if}\; \vert\tau\vert \le... ...ert^{-1}\tau} \,, &\text{if}\; \vert\tau\vert>1 \,. \end{cases}$$ (17)

Feeding this into the second equation in (13), we obtain the nonlinear variational equation

$$\nabla \nabla \varphi \varphi \forall \varphi \in$$ (18)

This problem is now discretized by the finite element method,

$$\nabla \nabla \varphi \varphi \forall \varphi \in$$ (19)

Combining (16) and (17), we obtain the nonlinear Galerkin orthogonality relation

$$\nabla \nabla \nabla \varphi \int_{0}^{1} \nabla \nabla \nabla \varphi$$ (20)

for  $\varphi$ $\in$ V_h , with the Jacobian  C'( ^. )  of the function  C( ^. ) , defined by

$$\tau \begin{cases} \quad I \,, &\text{if}\; \vert\tau\vert \le 1 ... ...tau^T \tau \} \,, &\text{if}\; \vert\tau\vert>1 \,. \end{cases}$$ (21)

Now, suppose that the quantity J(u) has to be computed. For representing the error  J(e) = J(u) - J(u_h), we use the solution z of the linear dual problem

$$\varphi \varphi \forall \varphi \in$$ (22)

with the bilinear form

$$\varphi \psi \int_{0}^{1} \nabla \nabla \varphi \nabla \psi$$

We assume that this dual solution is well defined. By the orthogonality relation (18), there holds

J(e) = L(u, u_h;e, z - z_h) ,

with a suitable approximation z_h $\in$ V_h. Then, analogously as in the linear case, we obtain the error representation

$$\sum_{K\in\mathbb T_h}^{} \left\{\vphantom{(f-\div C(\nabla u_h),z-z_h)_K - \textstyle{\frac{1}{2}}(n\cdot [C(\nabla u_h)],z-z_h)_{\partial K} }\right. \nabla {\frac{1}{2}} \nabla \partial \left.\vphantom{(f-\div C(\nabla u_h),z-z_h)_K - \textstyle{\frac{1}{2}}(n\cdot [C(\nabla u_h)],z-z_h)_{\partial K} }\right\}$$

From that, the following a posteriori error estimate follows,

$$\le \sum_{K\in\mathbb T_h}^{} \omega_{K}^{} \rho_{K}^{}$$ (23)

with the local residuals and weights defined by

$$\rho_{K}^{} \nabla \nabla \partial$$

$$\omega_{K}^{} \Big\{ \partial \Big\}$$

In order to evaluate these bounds, we replace the unknown solution  u  in the bilinear form L(u, u_h; ^. , ^. ) by the currently computed approximation  u_h , and solve the corresponding perturbed dual problem by the same method as used in computing  u_h , yielding an approximation $\tilde{z}_{h}^{}$ $\in$ V_h to the exact dual solution z,

$$\varphi \tilde{z}_{h}^{} \varphi \forall \varphi \in$$ (24)

The weights $\omega_{K}^{}$ may then again be approximated as in the linear case described above. We emphasize that the computation of the weights requires only to solve linear problems and normally only amounts to a small fraction of the total cost within a Newton iteration for the nonlinear problem.