The new adaptive finite element scheme

Our numerical scheme for solving the Prandtl-Reuss model in perfect plasticity (23) is based on the primal formulation (30) within an incremental loading process. In each load step a stationary Hencky-type problem is solved using an adaptive finite element discretization. The finite element meshes are optimized separately within each load step in accordance to the particular target functional  J( ^. )  leading to a dynamic development of refinement and coarsening over the whole calculation cycle.

We recall the notation for this framework. The space of admissible deformations is

V : = {$$\varphi$$ $$\in$$ H¹($$\Omega$$)²,  $$\varphi$$_{|$\Gamma_{u}$} = 0} .

Let  V_h $\subset$ V  be finite element spaces as described above, i.e., the underlying meshes consisting of triangles or quadrilaterals are shape-regular, but may contain hanging nodes, and the shape functions are linear or (isoparametricly) bilinear. Simplifying notation, the domain $\Omega$ is assumed to be polyhedral in order to ease the approximation of the boundary. More general situations may be treated by the usual modifications.

The loading process is started from the initial state  $\sigma^{0}_{}$ $\equiv$ 0. In each load step  t_{n - 1} $\rightarrow$ t_n , the nonlinear problem (30) has to be solved resulting in the following equation for the discrete deformation velocity  vⁿ_h $\in$ V_h,

$$\Pi \sigma^{n-1}_{h} \epsilon \epsilon \varphi \varphi \varphi \Gamma_{\sigma} \forall \varphi \in$$ (42)

From this, we obtain the stress update as

$$\sigma^{n}_{h} \Pi \sigma^{n-1}_{h} \epsilon$$ (43)

where the projection $\Pi$ is as defined above. Further, starting from the initial deformation u⁰_h $\equiv$ 0, we obtain corresponding approximate deformations as

uⁿ_h : = u^{n - 1}_h + k_nvⁿ_h .

Hence, (34) may be rewritten as

$$\Pi \sigma^{n-1}_{h} \epsilon \epsilon \varphi \varphi \varphi \Gamma_{\sigma} \forall \varphi \in$$ (44)

From now on, we suppress the sub- and superscript n and consider Hencky-type problems of the form

$$\Pi \hat{\sigma } \epsilon \hat{u} \epsilon \varphi \varphi \varphi \Gamma_{\sigma} \forall \varphi \in$$ (45)

with given initial stress  $\hat{\sigma }$  and deformation  $\hat{u}$ . From (37), we obtain deformations  u_h $\in$ V_h  and associated stresses

 $$\sigma_{h}^{}$$ : = $$\Pi$$($$\hat{\sigma }$$ + C$$\epsilon$$(u_h) - C$$\hat{u}$$) ,

approximating the solutions  u $\in$ V  and $\sigma$ $\in$ $\Pi$L²($\Omega$)^{d x d}_sym  of the corresponding continuous problem

$$\Pi \hat{\sigma } \epsilon \hat{u} \epsilon \varphi \varphi \varphi \Gamma_{\sigma} \forall \varphi \in$$ (46)

Clearly, for  $\hat{\sigma }$ = 0  and  $\hat{u}$ = 0, this reduces to the usual Hencky model (27).