2. Benchmark solution#

2.1. Calculation method#

It is an analytical solution.

Since the interface is adiabatic, there are two thermally insulated solids on either side of the interface. Each solid has a temperature imposed on part of its edge, and a zero flow on the rest of its edge. Since the problem is linear and stationary, the temperature is therefore constant in each part of the bar (and equal to the corresponding imposed temperature), and discontinuous across the interface:

in 3D: \(\{\begin{array}{c}T(x,y,z)=\text{}\stackrel{̄}{T}{\text{}}^{\text{inf}}=10°C,\text{}\forall \text{}(x,y,z)\text{}\in \text{}\left[\frac{-\mathit{Lx}}{2},\frac{\mathit{Lx}}{2}\right]\times \left[\frac{-\mathit{Ly}}{2},\frac{\mathit{Ly}}{2}\right]\times \left[\frac{-\mathit{Lz}}{2}\mathrm{,0}\right[\\ T(x,y,z)=\text{}\stackrel{̄}{T}{\text{}}^{\text{sup}}=20°C,\text{}\forall \text{}(x,y,z)\text{}\in \text{}\left[\frac{-\mathit{Lx}}{2},\frac{\mathit{Lx}}{2}\right]\times \left[\frac{-\mathit{Ly}}{2},\frac{\mathit{Ly}}{2}\right]\times \left]\mathrm{0,}\frac{\mathit{Lz}}{2}\right]\end{array}\)

in 2D: \(\{\begin{array}{c}T(x,y)=\text{}\stackrel{̄}{T}{\text{}}^{\text{inf}}=10°C,\text{}\forall \text{}(x,y)\text{}\in \text{}\left[\frac{-\mathit{Lx}}{2},\frac{\mathit{Lx}}{2}\right]\times \left[\frac{-\mathit{Ly}}{2}\mathrm{,0}\right[\\ T(x,y)=\text{}\stackrel{̄}{T}{\text{}}^{\text{sup}}=20°C,\text{}\forall \text{}(x,y)\text{}\in \text{}\left[\frac{-\mathit{Lx}}{2},\frac{\mathit{Lx}}{2}\right]\times \left]0,\frac{\mathit{Ly}}{2}\right]\end{array}\)

In the models presented in the following paragraphs, the approximation of the temperature field is enriched by a Heaviside function in order to represent the discontinuity introduced. The nodes whose support is crossed by the interface carry « classical » and « Heaviside » degrees of freedom: their values cannot then be directly compared to the analytical values obtained above.

In order to be able to test the values of these degrees of freedom, we consider the case where the mesh of the bar consists of 5 regular hexahedra with side \(\mathrm{1m}\), the central mesh is therefore cut in its middle by the interface. Since the solution is constant in the directions \(x\) and \(y\), we can reduce ourselves to considering the equivalent one-dimensional element represented in FIG.

_images/10000000000000B1000001081D60EC30015ED6DA.png

Figure 2.1-1: Equivalent 1D linear element

We note \(\text{N1}\) and \(\text{N2}\) the two nodes of this element, \({\phi }_{1}\) and \({\varphi }_{2}\) the associated shape functions, \(({T}_{1}^{C},{T}_{1}^{H})\) and \(({T}_{2}^{C},{T}_{2}^{H})\) the pairs of associated degrees of freedom. We also note \(x\) the space variable, \(\text{A}\) the point located in \(x={0}^{\text{+}}\) and \(\text{B}\) the point located in \(x\mathrm{=}{0}^{\text{-}}\). The Heaviside function characterizes either domain \(x<0\) (for node \(\text{N1}\)) or domain \(x>0\) (for node \(\text{N2}\)). Let us then note \({\chi }_{\text{-}}\) the characteristic function of the domain \(x<0\) and \({\chi }_{\text{+}}\) the characteristic function of the domain \(x>0\).

At any point \(x\) of the element, the temperature field is expressed by the following relationship:

\(T(x)={\phi }_{1}(x){T}_{1}^{C}+{\phi }_{2}(x){T}_{2}^{C}-2{\chi }_{\text{-}}(x){\phi }_{1}(x){T}_{1}^{H}+2{\chi }_{\text{+}}(x){\phi }_{2}(x){T}_{2}^{H}\)

note: the « -2 » and « +2 » multiplier coefficients are introduced to respect the convention for the definition of enrichment (see [R7.02.12]).

At points \(\text{N1}\), \(\text{N2}\), \(\text{A}\) and \(\text{B}\) we have:

\(\{\begin{array}{c}\text{En N1 :}{\chi }_{\text{+}}(x)=1,{\chi }_{\text{-}}(x)=0,{\phi }_{1}(x)=1,{\phi }_{2}(x)=0\\ \text{En A :}{\chi }_{\text{+}}(x)=1,{\chi }_{\text{-}}(x)=0,{\phi }_{1}(x)={\phi }_{2}(x)=1/2\\ \text{En B :}{\chi }_{\text{+}}(x)=0,{\chi }_{\text{-}}(x)=1,{\phi }_{1}(x)={\phi }_{2}(x)=1/2\\ \text{En N2 :}{\chi }_{\text{+}}(x)=0,{\chi }_{\text{-}}(x)=1,{\phi }_{1}(x)=0,{\phi }_{2}(x)=1\end{array}\)

which leads to the linear system:

\(\{\begin{array}{c}{T}_{1}^{C}=\text{}\stackrel{̄}{T}{\text{}}^{\text{sup}}\text{(N1)}\\ \frac{1}{2}{T}_{1}^{C}+\frac{1}{2}{T}_{2}^{C}+{T}_{2}^{H}=\text{}\stackrel{̄}{T}{\text{}}^{\text{sup}}\text{(A)}\\ \frac{1}{2}{T}_{1}^{C}-{T}_{1}^{H}+\frac{1}{2}{T}_{2}^{C}=\text{}\stackrel{̄}{T}{\text{}}^{\text{inf}}\text{(B)}\\ {T}_{2}^{C}=\text{}\stackrel{̄}{T}{\text{}}^{\text{inf}}\text{(N2)}\end{array}\)

assuming the following solution:

\(\{\begin{array}{c}{T}_{1}^{C}=\text{}\stackrel{̄}{T}{\text{}}^{\text{sup}}\\ {T}_{2}^{C}=\text{}\stackrel{̄}{T}{\text{}}^{\text{inf}}\\ {T}_{1}^{H}={T}_{2}^{H}=\frac{\text{}\stackrel{̄}{T}{\text{}}^{\text{sup}}-\text{}\stackrel{̄}{T}{\text{}}^{\text{inf}}}{2}\end{array}\),

whose digital application is equivalent to imposing: \(\{\begin{array}{c}{T}_{1}^{C}=20°C\\ {T}_{2}^{C}=10°C\\ {T}_{1}^{H}={T}_{2}^{H}=5°C\end{array}\)

2.2. Reference quantities and results#

We first test the values of the classical degrees of freedom TEMP and Heaviside H1 (noted respectively \({T}^{C}\) and \({T}^{H}\) in paragraph 2.1) of the temperature field at the output of the THER_LINEAIRE [U4.54.01] operator, at the nodes located just below and above the interface. Make sure to find the values determined in paragraph 2.1.

Identification

Reference

TEMPpour all nodes located just above the interface

\(20.°C\)

TEMPpour all nodes located just below the interface

\(10.°C\)

H1for all nodes located just below/above the interface

\(5.°C\)

The POST_MAIL_XFEM [U4.82.21] operator makes it possible to mesh the cracks represented by the X- FEM method. The operator POST_CHAM_XFEM [U4.82.22], then makes it possible to export the X- FEM results to this new mesh. These two operators should only be used after the calculation in post-processing views. They make it possible to generate nodes just below and above the interface and to show the field of nodal unknowns (here the temperature field). The value of the temperature field TEMP (noted \(T\) in paragraph 2.1) is then tested at the output of POST_CHAM_XFEM, at the nodes located just below and above the interface. Make sure to find the values determined in paragraph 2.1.

Identification

Reference

TEMPpour all nodes located just below the interface

\(10.°C\)

TEMPpour all nodes located just above the interface

\(20.°C\)

Finally, the value of the single component TEMP of the field TEMP_ELGA (temperature field by elements at the Gauss points, calculated only on the elements X- FEM by the operator THER_LINEAIRE) is tested on the Gauss points of the enriched elements.

Identification

Reference

TEMPsur the Gauss points located below the interface

\(10.°C\)

TEMPsur the Gauss points located above the interface

\(20.°C\)

Note: The X- FEM elements contain a large number of integration points (up to 480 for thermal elements based on the HEXA8 mesh) due to the division into sub-elements. Moreover, as the position of the crack determines the result of the sub-cutting procedure, this number is variable for the same type of element. We therefore do not test the value of the component TEMPdu field TEMP_ELGA on all the integration points of the elements X- FEM, and we just test it:

  • in a single integration point for enriched elements that are not crossed by the interface (the value being constant on the element)

  • in two integration points for the enriched elements crossed by the interface (the first located below the interface, the second above), and this only for models A, F, and I (mesh set in HEXA8ou * QUAD4)