6. Mortar Methods for Volume Coupled Problems 153
6.2.6. Conservation properties of the projections
In the following, some fundamental conservation properties for the projection operators are ana-lyzed. First, the standard patch test is reviewed. Second, weak conservation of nodal information during mesh transfer is investigated with respect to the employed integration domain.
6.2.6.1. Patch test
The scenario where a continuous field s can be exactly represented by the two considered meshesΩ1 andΩ2is considered, i.e.
s=N1s1 =N2s2. (6.26)
Then, the patch test is considered to be fulfilled if the projection operator introduces no errors, see Zienkiewicz and Taylor [306].
Since the collocation method is based on a node-wise information tracking as stated in (6.13), the patch test is automatically fulfilled when considering the assumed exact representation of fieldsby both meshes.
For the mortar method, this reads Pm21s1 =D−12 M21s1 =D−12 Thus, the mortar method also passes the patch test. Note, that this fact is completely independent from the chosen integration procedure, see Section 6.2.4.
6.2.6.2. Weak conservation of nodal information
Weak conservation of nodal information is fulfilled if the integral value of a continuous scalar field can be identically reproduced on the two considered meshesΩ1andΩ2, via
Z
Following Equations (6.3) and (6.4), the mortar approach reads This equation is valid for any discrete vectord2. Therefore, the equation reduces to
Z
The shape functions Φ2 of the dual field d2 fulfill the partition of unity requirement, which guarantees an unchanged integral value. Therefore, weak conservation of nodal information is automatically fulfilled in the mortar formulation for integration over the target meshΩ2, i.e.
Z
Note, however, that for curved boundaries and thus different volumes the weak integral conser-vation is not completely fulfilled for both domains in general, meaning that
Z
The collocation method can be interpreted as a degeneration of the mortar method. Here, the shape functions of the dual field become Dirac functions Φ2 = δ2, being infinity at the corresponding nodes and zero at all other points, which leads to an integral value of 1. Therefore, the integral definition in (6.30) reduces to
I2s2 =N˜21s1, (6.33)
where I2 is the identity matrix and N˜21 is the collocation matrix. Since the Dirac functions do not fulfill the partition of unity requirement, weak conservation of nodal information is never guaranteed for the collocation approach.
6.2.7. Numerical examples
In the following, the conservation properties of the developed mortar projection operator based on dual shape functions and the collocation projection operator are tested with two examples.
First, a classical structural patch test is employed in order to validate the abstract patch test requirement for projection operators as explained in Section 6.2.6.1. Second, the ability of weak conservation of nodal information is validated based on the investigations in Section 6.2.6.2.
Therefore, thermal energy is mapped from a source onto a target mesh of a cylindrical body.
Thus, also the boundary problems discussed in Section 6.2.5 are considered.
6.2.7.1. Conistency – Patch Test
As a first example, a purely structural patch test is considered. The idea of this first validation setup is to calculate a displacement solution on a first mesh, transfer the displacement field to
6.2. Fundamentals on volume projection of nodal information
Figure 6.7: 3D structural patch test for combinations of 8-node hexahedral, 20-node hexahedral, 27-node hexahedral, 4-node tetrahedral and 10-node tetrahedral elements. Left sub-figure: computed displacement solution on left mesh and projected displacement field on right mesh. Right subfigure: post-processed stress state for both meshes. Figure taken from Farah et al. [70].
a target mesh representing the same geometry and finally evaluate the resulting stress states on both meshes.
The geometrical setting is a cuboid of dimensions 4×4×16that is supported at its lower surface such that it is fixed in all directions. A constant pressure load ofp = 10,000is applied at the upper surface of the brick, thus leading to a uniaxially stretched deformation state. The material is modeled with a Saint-Venant-Kirchhoff law with Young’s modulus E = 210,000 and Poisson’s ratio ν = 0.0. For both meshes, arbitrary combinations of first-order as well as second-order finite element patches are employed, namely 8-node hexahedral, 20-node hexa-hedral, 27-node hexahexa-hedral, 4-node tetrahedral and 10-node tetrahedral elements. The different patches are connected by mortar mesh tying interfaces. Thus it is related to the structural patch test in Section 4.9.1.1. The first mesh is horizontally divided by several mesh tying interfaces, and the second mesh is vertically divided. This rather academic setup is chosen as challeng-ing as possible, i.e. requirchalleng-ing projections between all involved element types. The projectors for the volumetric mapping are created by the element-based integration scheme and the interface mortar mesh tying is evaluated by a segment-based integration scheme.
The resulting linear displacement field and the corresponding constant stress state on the first mesh are accurate up to machine precision. This is a characteristic feature of the interface mortar mesh tying method, see Puso [216] and the example in Section 4.9.1.1. The volumetric mapping of the displacement field onto the second mesh is done by employing the collocation and mor-tar projectors. Both projection methods are able to exactly transfer the displacement field, thus leading to the same constant stress state, see Figure 6.7. This result demonstrates that if a solu-tion can be exactly represented on both meshes, the collocasolu-tion approach as well as the mortar
α
Figure 6.8: Employed fine meshes for the source mesh (left) and the target mesh (right) for a rotation angle ofα= 30◦. Figure taken from Farah et al. [70].
approach do not introduce errors due to the mapping. Thus, as stated in the previous sections, both methods successfully pass the patch test for volume coupling.
6.2.7.2. Weak conservation – Thermal energy
In the following example, conservation properties of the projection methods are discussed with the help of another pure mesh transfer problem. The quantity to map can be interpreted as tem-perature field and the conservation property tested will be the thermal energy
Eθ =
Z
ΩCVθdΩ, (6.34)
with a constant heat capacity CV. The problem setting includes a cylindrical body, discretized with two different meshes, see Figure 6.8. On the source mesh, a given temperature fieldθ(r) =ˆ 20·(1−e0.1r2), that depends on the radial coordinate r of the cylinder will be approximated.
Note that the magnitude of θˆis deliberately chosen to have its highest value at the boundary of the cylinder in order to make the setup more challenging and to validate the boundary problems described in Section 6.2.5.2. The temperature field is projected onto the target mesh, where the thermal energy is re-evaluated. Thus, the error of the thermal energy on the target mesh with respect to the thermal energy on the source mesh is used as a measure for the global projection error. The target mesh is simply obtained by rotating the source mesh by an angleαaround the cylinder axis. In Figure 6.9, the results of the mapping are depicted exemplarily forα= 15◦. The collocation method was used for the evaluation of the projection operator. Another simulation with the mortar approach gives nearly indistinguishable results. The relative energy error over the rotation angle is depicted for first-order elements (8-node hexahedra) in Figure 6.10. Therein, the mortar and the collocation approach for a coarse mesh with22elements (left) and a finer mesh with760elements (right) are compared. In case of a zero rotation angle, both variants of course yield a perfect result, since the matching case is reproduced. Also, it becomes clear that the
6.2. Fundamentals on volume projection of nodal information
Figure 6.9: Temperature field on source mesh (left) and the target mesh (right) for a rotation an-gle ofα= 15◦. The collocation method was used for the evaluation of the projection operator. Another simulation with the mortar approach gives nearly indistinguishable results. Figure taken from Farah et al. [70].
coarser the mesh the higher the error, which is due to the non-matching boundary discretizations.
Still, the obtained error is small in general. Comparing the two approaches, one can conclude that for all rotation angles other than zero, the mortar approach gives far better results than the collocation method. In addition, the accuracy of the mapping is investigated with a coarse mesh employing second-order elements (27-node hexahedra). As shown in Figure 6.11, the overall relative error of the thermal energy decreases even further as compared with first-order elements.
However, the mortar method still yields far better results than the collocation approach.
Finally, the relative local errors of the thermal energy compared to the given temperature field are investigated. The results for the collocation method and the mortar method are shown in Figures 6.12 and 6.13, respectively. It can be seen that the collocation method yields an increased error on the target mesh. In contrast, the mortar method conserves the relative error and thus the discrete thermal energy. These results confirm the analysis of the global energy already presented in Figure 6.10 and demonstrate that the mortar projection operator performs excellently when weak conservation properties are required.
-0.5
Figure 6.10: Relative error of the thermal energy compared to reference meshes with 8-node hexahedral elements. Results for coarse mesh (left) and finer mesh (right). Visual-ization for mortar and collocation approach. Figure taken from Farah et al. [70].
-0.02
Figure 6.11: Relative error of the thermal energy compared to reference mesh with 27-node hex-ahedral elements. Results for mortar and collocation approach on a coarse mesh.
Figure taken from Farah et al. [70].
6.2. Fundamentals on volume projection of nodal information
Figure 6.12: Relative L2-error of thermal energy compared to analytical solution evaluated on each element for collocation method. Source mesh on the left side, target mesh on the right side. Results are shown forα= 15◦. Figure taken from Farah et al. [70].
Figure 6.13: Relative L2-error of thermal energy compared to analytical solution evaluated on each element for mortar method. Source mesh on the left side, target mesh on the right side. Results are shown forα = 15◦. Figure taken from Farah et al. [70].