5. Summary of the theoretical study#
Let (P0) be the mixed boundary problem (of the Cauchy-Dirichlet-Neumann-Robin type, inhomogeneous, linear and with variable coefficients) solved by the operator **** THER_LINEAIRE **
\(({P}_{0})\{\begin{array}{c}\rho {C}_{p}\frac{\partial T}{\partial t}-\text{div}(\lambda \nabla T)=s\Omega \times ]\mathrm{0,}\tau [\\ T=f{\mathrm{\Gamma }}_{1}\times ]\mathrm{0,}\tau [\\ \lambda \frac{\partial T}{\partial n}=g{\mathrm{\Gamma }}_{2}\times ]\mathrm{0,}\tau [\\ \lambda \frac{\partial T}{\partial n}+\text{hT}={\text{hT}}_{\text{ext}}{\mathrm{\Gamma }}_{3}\times ]\mathrm{0,}\tau [\\ T(x\mathrm{,0})={T}^{0}(x)\Omega \end{array}\) eq 5-1
Taking into account the choices of models made in Code_Aster (by AFFE_MATERIAU, AFFE_CHAR_THER…) we determine the minimum Variational Abstract Framework (CVA cf. [§2]) on which we will be able to rely to show the existence and the uniqueness of a solution temperature field (cf. [§2]). By crossing these somewhat « ethereal » theoretical prerequisites with the practical constraints of users, we deduce limitations as to the types of geometry and legal loads.
Then, while semi-discretizing in time and space using the usual methods of the code (which we of course ensure the validity of and the fact that they maintain the existence and uniqueness of the solution), we study the evolution of the stability properties of the problem (cf. [§3]). These controllability results are very useful for us to identify the norms, techniques and inequalities that are involved in the genesis of the residual indicator. In these discretization steps, we also briefly discuss the influence of this or that theoretical hypothesis on the functional perimeter of code operators.
Before summarizing the main theoretical results concerning the error indicator, we will respecify some notations:
we set a time step :math:`Deltat` such that :math:`frac{tau }{Deltat }` is an integer*N and that the time discretization is regular: \({t}_{0}=\mathrm{0,}{t}_{1}=\Deltat ,{t}_{2}=2\Deltat \cdots {t}_{n}=n\Deltat\),
Note:
This assumption of regularity is not really important, it just makes it possible to simplify the writing of the semi-discretized problem. To model any transient at time \({t}_{n}\), simply replace \(\Delta t\) with \(\Delta {t}_{n}={t}_{n}+1-{t}_{\mathrm{n.}}\)
let \(\theta\) be the parameter of the \(\theta\) -temporally semi-discretizing method \(({P}_{0})\),
let \({T}^{n}\) and \({T}_{h}^{n}\) be the temperature fields at time \({t}_{n}(0\le n\le N)\), exact solutions of the initial problem \(({P}_{0})\), respectively semi-discretized in time and completely discretized in time and space.
Given the models implemented in the code, we can assume that the time discretization of the loads and the source is correct and that the taking into account, via the Lagranges, of the Dirichlet boundary conditions (generalized or not) (generalized or not) is as well. On the other hand, one of the approaches to model the numerical approximations made during the integral calculations of the error indicator consists in assuming inaccurate the spatial discretization of the loads and the source. Their approximate values are noted.
\({s}_{\theta ,h}^{n\text{+}1},{g}_{\theta ,h}^{n\text{+}1},{T}_{\text{ext},\theta ,h}^{n\text{+}1}\text{et}{h}_{\theta ,h}^{n\text{+}1}\) eq 5-2
By posing
\({\chi }_{\theta }^{n\text{+}1}=\theta \chi (x,(n+1)\frac{\tau }{\Delta t})+(1-\theta )\chi (x,n\frac{\tau }{\Delta t})\text{avec}\chi \in \left\{T,s,{T}_{\text{ext}},g,h\right\}\text{et}0\le n\le N-1\) eq 5-3
Note:
The implementation of this type of indicator (in mechanics as well as in thermal) is also affected by another type of numerical approximations linked to the calculations of the second derivatives of the volume term (cf. [§4.3]). Its effect can possibly be felt when one is interested in the intrinsic value of the volume error for sources that are very chaotic on a coarse mesh.
There are then two constants K 2 and K 3 independent of the discretization parameters in time and space, depending only on the smallest angle of the triangulation and on the type of problem, which make it possible to build:
A**increase in the global spatial error**(the history of the global real indicator* « overestimate » ** the global spatial error)
\(\begin{array}{}{\parallel \sqrt{\rho {C}_{p}}({T}^{n}-{T}_{h}^{n})\parallel }_{\mathrm{0,}\Omega }^{2}+\Delta t\sum _{m=0}^{n\text{-}1}{\parallel \sqrt{\lambda }\nabla ({T}_{\theta }^{m\text{+}1}-{T}_{\theta ,h}^{m\text{+}1})\parallel }_{\mathrm{0,}\Omega }^{2}\\ \forall 0\le n\le N\le {\parallel \sqrt{\rho {C}_{p}}({T}_{0}-{T}_{0}^{h})\parallel }_{\mathrm{0,}\Omega }^{2}+{K}_{2}\Delta t\sum _{K\in {T}_{h}}{({\eta }_{R}^{0}(K))}^{2}+\sum _{m=0}^{n\text{-}1}\left\{{({\eta }_{R}^{m\text{+}1}(K))}^{2}+{h}_{K}^{2}{\parallel {s}_{\theta ,h}^{m\text{+}1}-{s}_{\theta }^{m\text{+}1}\parallel }_{\mathrm{0,}K}^{2}\right\}+\\ {K}_{2}\Delta t\sum _{K\in {T}_{h}}\sum _{m=0}^{n\text{-}1}\left\{\sum _{F\in {S}_{2}(K)}{h}_{F}{\parallel {g}_{\theta ,h}^{m\text{+}1}-{g}_{\theta }^{m\text{+}1}\parallel }_{\mathrm{0,}F}^{2}+\sum _{F\in {S}_{3}(K)}{h}_{F}{\parallel {({h}_{h}({T}_{\text{ext},h}-{T}_{h}))}_{\theta }^{m\text{+}1}-{(h({T}_{\text{ext}}-{T}_{h}))}_{\theta }^{m\text{+}1}\parallel }_{\mathrm{0,}F}^{2}\right\}\\ \end{array}\) eq 5-4
A**reduction in the local spatial error**(it « **underestimates* » the local spatial error)
\(\begin{array}{}{\eta }_{R}^{n\text{+}1}(K)\le {K}_{3}\left\{\begin{array}{}{h}_{K}{\parallel \sqrt{\rho {C}_{p}}\frac{{T}^{n\text{+}1}-{T}_{h}^{n\text{+}1}-{T}^{n}-{T}_{h}^{n}}{\Delta t}\parallel }_{\mathrm{0,}{\Delta }_{K}}+{\parallel \sqrt{\lambda }\nabla ({T}_{\theta }^{n\text{+}1}-{T}_{\theta ,h}^{n\text{+}1})\parallel }_{\mathrm{0,}{\Delta }_{K}}+\\ {h}_{K}{\parallel {s}_{\theta }^{n\text{+}1}-{s}_{\theta ,h}^{n\text{+}1}\parallel }_{\mathrm{0,}{\Delta }_{K}}+{h}_{F}^{\frac{1}{2}}{\parallel {g}_{\theta }^{n\text{+}1}-{g}_{\theta ,h}^{n\text{+}1}\parallel }_{\mathrm{0,}{\Delta }_{K}\cap {\Gamma }_{2}}+\\ {h}_{F}^{\frac{1}{2}}{\parallel {h}_{\theta }^{n\text{+}1}({T}_{\text{ext},\theta }^{n\text{+}1}-{T}_{\theta }^{n\text{+}1})-{h}_{\theta ,h}^{n\text{+}1}({T}_{\text{ext},\theta ,h}^{n\text{+}1}-{T}_{\theta ,h}^{n\text{+}1})\parallel }_{\mathrm{0,}{\Delta }_{K}\cap {\Gamma }_{3}}\end{array}\right\}\\ \forall 0\le n\le N-1\end{array}\) eq 5-5
With**the sequence*\({({\eta }_{R}^{n}(K))}_{0\le n\le N}^{K\in {T}_{h}}`**of real local indicators** (using [:ref:`§4.1 <§4.1>\)] ratings)
\(\begin{array}{}{\eta }_{R}^{n\text{+}1}(K):={\eta }_{R,\text{vol}}^{n\text{+}1}(K)+{\eta }_{R,\text{saut}}^{n\text{+}1}(K)+{\eta }_{R,\text{flux}}^{n\text{+}1}(K)+{\eta }_{R,\text{éch}}^{n\text{+}1}(K)\\ :={h}_{K}{\parallel {s}_{\theta ,h}^{n\text{+}1}-\rho {C}_{p}\frac{{T}_{h}^{n\text{+}1}-{T}_{h}^{n}}{\Delta t}+\text{div}(\lambda \nabla {T}_{h,\theta }^{n\text{+}1})\parallel }_{\mathrm{0,}K}+\frac{1}{2}\sum _{F\in {S}_{\Omega }(K)}\sqrt{{h}_{F}}{\parallel \left[\lambda \frac{\partial {T}_{h,\theta }^{n\text{+}1}}{\partial n}\right]\parallel }_{\mathrm{0,}F}+\\ \sum _{F\in {S}_{2}(K)}\sqrt{{h}_{F}}{\parallel {g}_{\theta ,h}^{n\text{+}1}-\lambda \frac{\partial {T}_{h,\theta }^{n\text{+}1}}{\partial n}\parallel }_{\mathrm{0,}F}+\sum _{F\in {S}_{3}(K)}\sqrt{{h}_{F}}{\parallel {(h({T}_{\text{ext}}-T))}_{\theta ,h}^{n\text{+}1}-\lambda \frac{\partial {T}_{h,\theta }^{n\text{+}1}}{\partial n}\parallel }_{\mathrm{0,}F}\end{array}\) eq 5-6
which is initialized by
\(\begin{array}{}{\eta }_{R}^{0}(K):={h}_{K}{\parallel {s}_{h}^{0}+\text{div}(\lambda \nabla {T}_{h}^{0})\parallel }_{\mathrm{0,}K}+\frac{1}{2}\sum _{F\in {S}_{\Omega }(K)}\sqrt{{h}_{F}}{\parallel \left[\lambda \frac{\partial {T}_{h}^{0}}{\partial n}\right]\parallel }_{\mathrm{0,}F}+\\ \sum _{F\in {S}_{2}(K)}\sqrt{{h}_{F}}{\parallel {g}_{h}^{0}-\lambda \frac{\partial {T}_{h}^{0}}{\partial n}\parallel }_{\mathrm{0,}F}+\sum _{F\in {S}_{3}(K)}\sqrt{{h}_{F}}{\parallel {(h({T}_{\text{ext}}-T))}_{h}^{0}-\lambda \frac{\partial {T}_{h}^{0}}{\partial n}\parallel }_{\mathrm{0,}F}\end{array}\) eq 5-7
This local suite makes it possible to build the suit \({({\eta }^{n}(\Omega ))}_{0\le n\le N}\) of global real indicators
\(\forall 0\le n\le N{\eta }_{R}^{n}(\Omega ):={(\sum _{K\in {T}_{h}}{\eta }_{R}^{n}{(K)}^{2})}^{\frac{1}{2}}\) eq 5-8
From [éq 5-4] (cf. [§4.2]) it appears that, at a given moment, the error in approximating the Cauchy condition and the history of the global indicators affects the overall quality of the solution obtained. We will therefore be able to globally minimize the approximation error due to finite elements over time by remelining the structure « wisely », via a series of indicators, the structure. Because, in practice, we see that the refinement of the meshes makes it possible to reduce their error and therefore to reduce the temporal sum of the indicators. The global error will (and this is moral) hit the floor value due to approximations of the initial condition, the limit conditions and the source (which will also tend to decrease of course!) . You can’t get better quality results than the input data for the problem!
The result [éq 5-5] (cf. [§4.4]) only provides a local inverse of the global increase [éq 5-4] (the « must » would have been to also show an increase at the local level) but, within the framework of this type of indicator, we will not be able to obtain a better compromise. These estimates are optimal in the sense of [bib5]. They illustrate the equivalence of the Hilbertian sum of the indicators with the spatial part of the global exact error. The equivalence constants are independent of the discretization parameters in space and time, they only depend on the smallest angle of the triangulation and on the type of problem treated.
According to this increase in the indicator [éq 5-6], if one refines very locally (around the element \(K\)) in order to reduce \({\eta }_{R}^{n}(K)\), one is not assured of a decrease in the error in the immediate vicinity of the zone concerned (in \({\Delta }_{K}\)). The indicator locally « underestimates » the spatial error and only a more macroscopic refinement theoretically achieves a decrease in the error.
As a pure residue alone, a whole « zoology » of spatial error indicators are legible (cf. [§4.3]), we have retained a type similar to the one already put in place for the mechanics of Code_Aster. Based on the solutions and the discrete loadings of the current moment and the previous moment (except at the first time step), its theoretical limitations are therefore, at best, those inherent in solving the temperature problem: no areas with brush or peak points, no crack, problem at multi-material interfaces, \(\theta\) -unconditionally stable scheme, regular triangulation family, mesh polygonal discretized by isoparametric finite elements, oscillations and violation of the principle of the maximum (cf. [§4.5]). Of course, in practice, this « theoretical » scope of use is often ignored, and without hindrance.
But it should be borne in mind that, as a « simple post-treatment » of \(({P}_{0})\) , the indicator unfortunately cannot provide a more reliable diagnosis in areas where the resolution of the initial problem is unsuccessful (near a crack, shock…). In these particular cases, its carefully reserved designation of indicator (instead of the usual terminology of estimator) is more appropriate than ever! But while, in these extreme cases, its raw value may be questionable, its usefulness as an efficient and convenient supplier of error cards for re-meshing or refining/deraffinating remains completely justified.
In the same vein, even if the formulation [éq 5-6] has only been established in the transient linear case, isotropic or not, defined by (\({P}_{0}\)), we could also stretch its perimeter of use to non-linear (operator THER_NON_LINE), to different limit conditions (ECHANGE_PAROI for example), to different boundary conditions (for example) or to other types of finite elements (lumped isoparametric elements, structural elements…) (cf. [§2.1]). For more information on the « computer » perimeter corresponding to its effective implementation in the code, we can refer to [§6.2] or to the user documentation of CALC_ERREUR [U4.81.06].
Hitherto, there has only been talk of maps of spatial error indicators calculated at a given moment of the calculation transient. But, in fact, there are several ways to build an error indicator on a parabolic problem (cf. [§4.5]). The one we selected does not allow a complete control of the error and it always requires some vigilance when treating shock-type problems (the same as for the post-treated problem!). It only makes the term time jump appear implicitly in all the terms in \(\theta\) of [éq 5-6].
Finally, it should be noted that this indicator is therefore composed of four terms:
the**main term*, called**volume error term**, controlling the correct verification of the heat equation,
to which are added**three secondary terms**verifying the good performance of**space jumps**and limit conditions:**terms of flow and exchange*.
In 2D- PLAN or in 3D (resp. in 2D- AXI), if the unit of geometry is the meter, the unit of the former is the W.m (resp. \(W\text{.}m\text{.}{\text{rad}}^{-1}\)) and that of the other terms is the \(W\text{.}{m}^{\frac{1}{2}}\) (resp. \(W\text{.}{m}^{\frac{1}{2}}\text{.}{\text{rad}}^{-1}\)). Attention therefore to the units taken into account for geometry when interested in the raw value of the indicator and not in its relative value!
After the practical difficulties of implementation in the code, we will now discuss the necessary environment and its scope of use. We will conclude with an example of use taken from an official test case.