A new mortar formulation for modeling elastomer bedded structures with modal-analysis in 3D
- Thomas Horger^{1}Email author,
- Stefan Kollmannsberger^{2},
- Felix Frischmann^{2},
- Ernst Rank^{2} and
- Barbara Wohlmuth^{1}
DOI: 10.1186/s40323-014-0018-0
© Horger et al.; licensee Springer. 2014
Received: 25 April 2014
Accepted: 30 July 2014
Published: 6 November 2014
Abstract
Background
It is a well-known fact that cross-laminated timber structures are sensitive to rumbling noises. These transmissions are best captured by a fully three-dimensional mathematical model. Since the discretization of such models with hexahedral elements in a conforming manner is highly complex, we chose the mortar method to reduce the algorithmic complexity for the mesh generation. Moreover we consider high-order finite elements in order to deal with the high aspect ratios in three-dimensionally resolved, cross-laminated walls and slabs. The geometric models and material specification was derived from a building information model.
Methods
This paper derives a new mortar formulation designed to replace an explicitely discretized elastomer with a new coupling condition. To this end, tailored Robin conditions are applied at the interface as coupling conditions instead of the more standard continuity constraints. Having demonstrated the suitability of the mortar method for high order finite elements, we proceed with the derivation of the dimensional reduced model with the new coupling condition and to show its stability by numerical experiments. We then test the performance of the new formulation on benchmark examples and demonstrate the engineering relevance for practical applications.
Results
The newly derived mortar formulation performs well. We tested the new formulation on fully three-dimensional examples of engineering relevance discretized by high-order finite elements up to degrees of p=10 and found the reproduction of both eigenvalues and eigenmodes to be accurate. Moreover, the mortar method allows for a significant reduction in the algorithmic complexity of mesh generation while simultaneously reducing the overall computational effort.
Conclusion
The newly derived modified mortar method for replacing an elastomer layer is not only an academically interesting variant but is capable of solving problems of practical importance in modal-analysis of cross-laminated timber structures.
Keywords
Mortar method Weak coupling High-order finite elements Eigenvalue problem Cross laminated timber structures Modal-analysisBackground
The main contribution of this paper is a new dimensionally reduced model which captures eigenvalues and eigenmodes of elastomeric coupled domains in timber structures. Dimensionally reduced models are very attractive from the computational point of view. There is no need to mesh the three dimensional but thin subdomain of the elastomer within our approach. However, new challenges arise such as the formulation of a suitable coupling condition and their numerical realization. Here we use a variant of the popular mortar finite element method [1]-[3]. Mortar methods can be analyzed within the abstract framework of saddlepoint problems and can be regarded as a domain decomposition technique. Firstly, coupled problems are teared, meshed and discretized separately resulting, in general, in non-matching meshes at the interfaces. Secondly, these independent subproblems are interconnected in a weak form by balance equations involving, e.g., the surface traction. Thus, these techniques provide a very flexible and computationally attractive setting to handle numerically coupled multi-physics problems. Mortar methods have been applied successfully in many engineering applications, such as, e.g., contact problems [4]-[7], dynamic and static structural analysis [8]-[10], flow problems [11]-[13] and coupled problems in acoustics [14],[15]. Further, the mortar method is used to simulate eigenvalue problems in [16],[17]. Most contributions deal only with first or second order approaches. Although the theory of high order mortar methods is well understood [18],[19], the implementation of higher order quadrature formulas on cut elements in 3D simulations is technical challenging. Here we apply high order, up to 20 in the polynomial degree, techniques to approximate eigenvalues and eigenmodes in cross laminated timber structures interconnected by thin elastomer structures.
Our motivation to derive such a formulation stems from the need to compute the modal-analysis which is a main part of vibro-acoustical-analysis. In order to control sound transmissions between slabs and walls, these components are often connected by elastomers which we firstly model by using the linear elasticity equation because of the very thin character. Due to the composition of timber constructions consisting of thin, layered and orthotropic material, we aim for a fully three-dimensional resolution of the slabs and walls. For this purpose, we use the p-version of the finite element method, as presented for example in [20]. Moreover, it is well suited for the computation of solid, but thin-walled structures because it is robust in terms of the large aspect ratios which arise naturally in fully three-dimensional models of plates and shells [21]. It also provides better accuracy and convergence properties than low-order finite elements. In addition, the p-version of the FEM has already been shown to lead to excellent results for the analysis of sound transition through timber floors [22].
However, the construction of conforming, three dimensional meshes, that are analysis-suitable, is non-trivial. In this paper, we utilize the mesh generation techniques presented in [23]. A conforming mesh of connected walls and slabs, increases the number of elements significantly, as a local mesh refinement, in only one of the components automatically spreads to the others.
These restrictions motivate the use of mortar methods allowing for an independent meshing of the individual building components, as the physically imperative coupling is carried out numerically at a later stage in a weak sense.
The mortar method was first introduced as a method to couple spectral elements with finite elements in [1] where the ansatz space was weakly constrained. The present contribution, however, views the mortar method in the more popular context of enforcing the coupling conditions by means of Lagrange multipliers, as introduced in [2], and thus resulting in a saddle point formulation.
The modeling of elastic interface boundary conditions has been the subject for low orders in [24]-[26]. Also the modeling of interface elements has been investigated in [27],[28], with a spring boundary condition in [29] and with a Robin-type condition in [30]. We built on the work of [31], which demonstrated the excellent applicability of the mortar method for problems in structural mechanics for discretizations of high orders. We extend this concept to elastomeric coupled domains. To this end, we enforce a non-standard Robin type condition at the interface by means of Lagrange multipliers instead of the continuity requirements. Robin type interface conditions have been used to glue nonconforming grids, see, e.g., [32]. The main difference to the current paper is that our coupling condition not only aims to glue two nonconforming grids together, but is also able to replace the whole explicit discretization of an elastomer. Therefore, it goes beyond a simple domain decompositon method, it provides also a dimensionally reduced model.
The contribution at hand is organized as follows: We start by presenting the problem setting in Section ‘Problem setting and conforming discretization’ and introduce the classical mortar method in Section ‘Mortar method’. In Section ‘Modeling of the elastomer’, we derive our new mortar coupling condition which is able to replace an explicitly discretized elastomer. In Section ‘Results and discussion’, we present our simulation results. Section ‘Results and discussion’ compares numerically the standard mortar method with the conforming high order method in the context of eigenvalue problems for a rigidly connected L-shaped wall-slab example. To establish a reference solution, we firstly compute the eigenvalues and eigenfunctions on a wall-slab configuration in a conforming discretization in Section ‘Results and discussion’. There we already investigate the effect of connecting walls and slabs with different elastomers on the eigenvalues and the eigenfunctions. We then test the new formulation on the same wall-slab configuration in Section ‘The new elastomeric coupled mortar formulation’. Section ‘Influence of the elastomer thickness’ analyses numerically the influence of the elastomer thickness on the new coupling condition. Furthermore a more complex and application relevant example is presented in Section ‘A complex example’. In Section ‘Conclusions’, we give some conclusion according to the numerical results showing the flexibility and robustness of the new mortar method for practical application.
Methods
Problem setting and conforming discretization
In this section, we provide a dimensionally reduced model, resulting in a modified mortar approach. In contrast to the classical mortar setting, we end up with a non-symmetric saddle-point formulation. The surface traction now enters as a spring into the coupling condition.
We discretize Equation (3) using conforming finite elements of high order associated with a hexahedral mesh. As basis functions, we use hierarchical shape functions based on integrated Legendre polynomials [20],[33].
Mortar method
The eigenvalue problem (1) can then be written in the following variational form:
Equation (5) now defines the saddle point problem arising from the mortar method. The Lagrange multiplier λ corresponds to the negative surface traction -σ n of Ω_{ s } on the interface Γ, where n is the outward unit normal of Ω_{ s }.
with A(u,λ;v,μ):=a(u,v)+b(v,λ)+b(u,μ).
The bilinearform A(•,•;•,•) fulfills the conditions of Remark 13.4 in [35], and thus the theory given in Section 8 of [35] ensures convergence of the discrete eigenvalues and eigenfunctions.
Modeling of the elastomer
The modeling of an elastomer for vibration isolation has been the subject in [36],[37]. These papers take many mechanical properties like strain and damping directly into account. Alternatively, the modal- and spectral-analysis can be realized by the modal superposition. In this case, the eigenmodes of the undamped system are required, and the damping is only taken into account in a postprocessing step. Thus, we neglect the damping. Moreover, the elastomer is modeled in terms of the linear elasticity equations because it is comparatively thin in one space direction [22]. This section will lay out a new modeling approach using a Robin type condition for the coupling, in order to replace an elastomer. This new coupling condition results in a dimensional reduced model which avoids the meshing of the three dimensional subdomain which corresponds to the elastomer. Our new coupling condition still yields a saddle point problem which fits into the implementational framework of mortar methods.
Modified mortar method using a Robin type condition
Equation (7) is the new coupling condition between displacements and surface traction in the strong form.
with the modeling parameter α defined as
Note that the parameters α and β can be directly computed from the properties of the elastomer. Replacing X by X_{ h } and M by M_{ h } gives the discrete version of Equation (9) yielding approximations ω_{ h } of the eigenvalues.
Results and discussion
Comparison between conforming and mortar discretization
Elastomer properties for the simulations
Timber | Elast 1 | Elast 2 | Elast 3 | Elast 4 | Elast 5 | |
---|---|---|---|---|---|---|
Young's-modulus in [ N/m^{2}] | 9790•10^{6} | 1.8•10^{7} | 8.0•10^{6} | 3.7•10^{6} | 1.7•10^{6} | 8.0•10^{5} |
Poisson v in [–] | 0.05 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 |
Comparison of eigenfrequency of the L-shaped wall-slab configuration
p = 3 | p = 7 | |||||
---|---|---|---|---|---|---|
EW | Conform | Mortar | % | Conform | Mortar | % |
1 | 50.720 | 50.852 | 0.261 | 50.289 | 50.298 | 0.019 |
2 | 70.755 | 72.006 | 1.768 | 69.172 | 69.942 | 1.113 |
3 | 76.534 | 78.317 | 2.330 | 74.456 | 74.833 | 0.506 |
4 | 90.707 | 91.976 | 1.399 | 87.931 | 88.491 | 0.637 |
5 | 159.423 | 168.390 | 5.624 | 125.276 | 126.069 | 0.632 |
6 | 174.712 | 174.869 | 0.090 | 159.311 | 159.393 | 0.051 |
7 | 179.359 | 185.147 | 3.227 | 172.931 | 172.966 | 0.020 |
p = 10 | p = 15 | |||||
EW | Conform | Mortar | % | Conform | Mortar | % |
1 | 50.282 | 50.288 | 0.012 | 50.278 | 50.281 | 0.006 |
2 | 68.929 | 69.518 | 0.854 | 68.749 | 69.062 | 0.455 |
3 | 74.304 | 74.535 | 0.311 | 74.220 | 74.341 | 0.162 |
4 | 87.685 | 88.101 | 0.475 | 87.545 | 87.768 | 0.255 |
5 | 124.818 | 125.340 | 0.418 | 124.581 | 124.842 | 0.210 |
6 | 159.264 | 159.315 | 0.032 | 159.237 | 159.264 | 0.016 |
7 | 172.884 | 172.911 | 0.016 | 172.865 | 172.882 | 0.010 |
Discrete modeling of the elastomer
Influence of the different elastomers on the eigenfrequencies given in [ H z ]
EW | No Elast. | Elast. 1 | Elast. 2 | Elast. 3 | Elast. 4 | Elast. 5 |
---|---|---|---|---|---|---|
1 | 50.282 | 48.584 | 47.472 | 45.933 | 43.157 | 38.357 |
2 | 68.929 | 52.437 | 51.461 | 50.461 | 48.676 | 45.275 |
3 | 74.304 | 64.128 | 61.773 | 58.287 | 52.669 | 45.588 |
4 | 87.685 | 79.851 | 77.797 | 74.245 | 68.109 | 59.885 |
5 | 124.818 | 110.669 | 105.449 | 98.276 | 90.290 | 84.003 |
6 | 159.264 | 149.448 | 141.577 | 127.098 | 106.626 | 89.151 |
7 | 172.884 | 160.956 | 154.662 | 140.762 | 123.733 | 105.596 |
8 | 178.886 | 162.633 | 155.910 | 145.873 | 127.320 | 111.518 |
The new elastomeric coupled mortar formulation
We now test the new mortar model given by Equation (9) using the discretization depicted on the right-hand side of Figure 4. The results are compared to the classical, conforming discretization, as depicted on the left-hand side of Figure 4, where the elastomer was modeled explicitly, as described in Section ‘Results and discussion’.
Eigenfrequencies given in [ H z ] for the new modeling approach along with the deviation in percent from the conforming discretization depicted on the left hand side of Figure 4
Elast 1 | Elast 2 | Elast 3 | Elast 4 | Elast 5 | ||||||
---|---|---|---|---|---|---|---|---|---|---|
EW | Value | % | Value | % | Value | % | Value | % | Value | % |
1 | 48.664 | 0.165 | 47.511 | 0.082 | 46.034 | 0.218 | 43.206 | 0.112 | 38.545 | 0.490 |
2 | 52.678 | 0.459 | 51.628 | 0.325 | 50.685 | 0.443 | 48.997 | 0.659 | 45.846 | 1.262 |
3 | 64.315 | 0.292 | 61.916 | 0.231 | 58.685 | 0.682 | 52.891 | 0.421 | 46.082 | 1.083 |
4 | 80.059 | 0.260 | 78.252 | 0.585 | 75.113 | 1.170 | 69.159 | 1.542 | 61.539 | 2.763 |
5 | 110.912 | 0.220 | 105.784 | 0.317 | 99.112 | 0.850 | 90.606 | 0.350 | 84.208 | 0.243 |
6 | 149.371 | 0.052 | 141.757 | 0.127 | 128.750 | 1.300 | 107.303 | 0.635 | 89.468 | 0.355 |
7 | 161.365 | 0.254 | 155.063 | 0.259 | 142.314 | 1.103 | 124.127 | 0.319 | 109.623 | 3.814 |
8 | 162.967 | 0.205 | 157.058 | 0.737 | 148.494 | 1.797 | 130.530 | 2.521 | 111.558 | 0.036 |
Modal assurance criterion for the modeling of Elastomer 5
u _{1} | u _{2} | u _{3} | u _{4} | u _{5} | u _{6} | u _{7} | u _{8} | ||
---|---|---|---|---|---|---|---|---|---|
u _{1} | 1.000 | 0.000 | 0.004 | 0.000 | 0.002 | 0.000 | 0.000 | 0.000 | |
u _{2} | 0.000 | 1.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | |
u _{3} | 0.003 | 0.000 | 0.999 | 0.000 | 0.000 | 0.000 | 0.003 | 0.000 | |
${\mathit{\text{MAC}}}_{E5\text{\_}1.2\left[\mathit{\text{cm}}\right]}$= | u _{4} | 0.000 | 0.000 | 0.000 | 0.999 | 0.000 | 0.000 | 0.000 | 0.000 |
u _{5} | 0.001 | 0.000 | 0.000 | 0.000 | 1.000 | 0.000 | 0.000 | 0.000 | |
u _{6} | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 1.000 | 0.002 | 0.000 | |
u _{7} | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.998 | 0.000 | |
u _{8} | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 1.000 |
Influence of the elastomer thickness
A key assumption of the new approach is that the displacement field varies only linearly in the direction perpendicular to the two opposite interface of the elastomer with adjacent structures. In order to investigate the validity of this assumption, we vary the thickness of the elastomer and show its influence on the corresponding eigenvalues.
Remark1.
At this point it is noted that the thickness of the elastomers for typical wall-slab configurations is below 3[ c m]. In practical applications, thicknesses range from 1[ c m] to 1.5[ c m].
Eigenfrequencies given in [ Hz ] for the conform and the new method with the corresponding deviation in [%] for the elastomer thickness 3[cm]
Elast 1 | Elast 3 | Elast 5 | |||||||
---|---|---|---|---|---|---|---|---|---|
EW | Conform | New | % diff | Conform | New | % diff | Conform | New | % diff |
method | method | method | method | method | method | ||||
1 | 46.873 | 46.828 | 0.096 | 42.043 | 42.282 | 0.568 | 29.716 | 30.419 | 2.367 |
2 | 51.223 | 51.504 | 0.549 | 47.783 | 48.655 | 1.825 | 36.608 | 37.718 | 3.033 |
3 | 60.827 | 61.081 | 0.416 | 50.929 | 51.658 | 1.432 | 37.274 | 39.155 | 5.047 |
4 | 76.991 | 78.028 | 1.347 | 65.466 | 68.104 | 4.031 | 48.848 | 51.360 | 5.143 |
5 | 103.857 | 104.722 | 0.833 | 88.359 | 89.365 | 1.139 | 76.357 | 76.166 | 0.251 |
6 | 138.595 | 139.275 | 0.491 | 101.474 | 103.915 | 2.406 | 79.253 | 79.477 | 0.283 |
7 | 151.783 | 152.988 | 0.794 | 119.491 | 121.222 | 1.449 | 84.601 | 88.684 | 4.826 |
8 | 153.680 | 156.466 | 1.813 | 119.571 | 127.398 | 6.546 | 103.182 | 102.752 | 0.416 |
Eigenfrequencies given in [ Hz ] for the conform and the new method with the corresponding deviation in [%] for the elastomer thickness 4[cm]
Elast 1 | Elast 3 | Elast 5 | |||||||
---|---|---|---|---|---|---|---|---|---|
EW | Conform | New | % diff | Conform | New | % diff | Conform | New | %diff |
method | method | method | method | method | method | ||||
1 | 46.100 | 46.001 | 0.215 | 39.931 | 40.471 | 1.354 | 26.683 | 27.634 | 3.562 |
2 | 50.780 | 51.115 | 0.661 | 46.091 | 47.664 | 3.413 | 32.869 | 35.531 | 8.098 |
3 | 59.252 | 59.736 | 0.817 | 47.753 | 48.835 | 2.266 | 35.046 | 36.449 | 4.004 |
4 | 75.331 | 76.804 | 1.956 | 61.409 | 65.291 | 6.323 | 45.903 | 48.787 | 6.283 |
5 | 100.661 | 102.266 | 1.594 | 85.356 | 86.606 | 1.464 | 73.943 | 73.419 | 0.707 |
6 | 132.275 | 134.570 | 1.735 | 93.278 | 96.202 | 3.135 | 78.124 | 78.372 | 0.317 |
7 | 145.499 | 148.266 | 1.902 | 108.759 | 115.437 | 6.140 | 80.538 | 84.663 | 5.122 |
8 | 148.833 | 153.236 | 2.959 | 113.584 | 119.391 | 5.113 | 101.426 | 100.758 | 0.658 |
Modal assurance criterion for the modeling of Elastomer 5 with thickness 3[cm]
u _{1} | u _{2} | u _{3} | u _{4} | u _{5} | u _{6} | u _{7} | u _{8} | ||
---|---|---|---|---|---|---|---|---|---|
u _{1} | 1.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.001 | 0.000 | 0.000 | |
u _{2} | 0.000 | 1.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | |
u _{3} | 0.000 | 0.000 | 0.996 | 0.000 | 0.000 | 0.000 | 0.003 | 0.000 | |
${\mathit{\text{MAC}}}_{E5\text{\_}3\mathit{\text{cm}}}$= | u _{4} | 0.000 | 0.000 | 0.000 | 0.999 | 0.000 | 0.000 | 0.000 | 0.000 |
u _{5} | 0.001 | 0.000 | 0.000 | 0.000 | 0.998 | 0.000 | 0.006 | 0.000 | |
u _{6} | 0.002 | 0.000 | 0.000 | 0.000 | 0.002 | 0.999 | 0.001 | 0.000 | |
u _{7} | 0.000 | 0.000 | 0.004 | 0.000 | 0.000 | 0.000 | 0.994 | 0.000 | |
u _{8} | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 1.000 |
Modal assurance criterion for the modeling of Elastomer 5 with thickness 4[cm]
u _{1} | u _{2} | u _{3} | u _{4} | u _{5} | u _{6} | u _{7} | u _{8} | ||
---|---|---|---|---|---|---|---|---|---|
u _{1} | 1.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.001 | 0.000 | 0.000 | |
u _{2} | 0.000 | 1.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | |
u _{3} | 0.000 | 0.000 | 0.994 | 0.000 | 0.000 | 0.000 | 0.003 | 0.000 | |
${\mathit{\text{MAC}}}_{E5\text{\_}4\mathit{\text{cm}}}$= | u _{4} | 0.000 | 0.000 | 0.000 | 0.999 | 0.000 | 0.000 | 0.000 | 0.000 |
u _{5} | 0.002 | 0.000 | 0.000 | 0.000 | 0.998 | 0.000 | 0.008 | 0.000 | |
u _{6} | 0.002 | 0.000 | 0.000 | 0.000 | 0.002 | 0.999 | 0.001 | 0.000 | |
u _{7} | 0.000 | 0.000 | 0.008 | 0.000 | 0.000 | 0.000 | 0.990 | 0.000 | |
u _{8} | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 1.000 |
A complex example
Layer thicknesses of walls and slab 125
Type | Layering [m m] |
---|---|
61 | 17*-27-17* |
85 | 17*-17-17*-17-17* |
95 | 17*-17-27*-17-17* |
125 | 27*-27-17*-27-27* |
Computed eigenfrequencies given in [ H z ]for the building example
EW | Conform | Mortar | % diff | Conform | New coupling | % diff |
---|---|---|---|---|---|---|
no elast | no elast | elast | elast | |||
1 | 11.357 | 11.471 | 1.007 | 9.883 | 9.960 | 0.779 |
2 | 13.738 | 13.861 | 0.899 | 12.439 | 12.496 | 0.459 |
3 | 14.347 | 14.425 | 0.547 | 13.302 | 13.346 | 0.330 |
4 | 15.807 | 15.947 | 0.884 | 13.938 | 14.067 | 0.926 |
5 | 16.988 | 17.133 | 0.856 | 14.980 | 15.134 | 1.030 |
6 | 21.070 | 21.329 | 1.227 | 19.256 | 19.398 | 0.737 |
7 | 21.832 | 21.988 | 0.715 | 20.765 | 20.833 | 0.325 |
8 | 24.038 | 24.265 | 0.947 | 21.072 | 21.165 | 0.437 |
... | ... | ... | ... | ... | ... | ... |
20 | 36.868 | 37.071 | 0.552 | 34.033 | 34.437 | 1.189 |
... | ... | ... | ... | ... | ... | ... |
30 | 48.414 | 48.769 | 0.732 | 43.329 | 43.850 | 1.202 |
... | ... | ... | ... | ... | ... | ... |
40 | 61.815 | 62.479 | 1.073 | 53.238 | 53.574 | 0.631 |
... | ... | ... | ... | ... | ... | ... |
50 | 69.224 | 70.028 | 1.162 | 60.897 | 61.468 | 0.938 |
... | ... | ... | ... | ... | ... | ... |
60 | 77.711 | 78.402 | 0.889 | 66.702 | 67.982 | 1.919 |
... | ... | ... | ... | ... | ... | ... |
70 | 86.225 | 86.443 | 0.253 | 76.123 | 76.488 | 0.479 |
... | ... | ... | ... | ... | ... | ... |
80 | 93.425 | 93.893 | 0.501 | 83.881 | 84.382 | 0.597 |
... | ... | ... | ... | ... | ... | ... |
90 | 101.063 | 101.673 | 0.603 | 88.875 | 89.558 | 0.769 |
... | ... | ... | ... | ... | ... | ... |
100 | 108.871 | 109.382 | 0.469 | 94.814 | 95.145 | 0.349 |
Conclusions
The aim of this contribution was to model the behavior of eigenvalue problems of elastomerically supported, cross-laminated timber structures by means of an extended mortar method.
To this end, we first evaluated the applicability of the mortar method to the p-version of the finite element method of an eigenvalue problem for three-dimensional shell and plate-like structures. The deviation from a conformingly discretized, stiffly coupled wall-slab configuration for higher order p is below 1[ %] for all investigated eigenvalues. The eigenmodes likewise provided an excellent match within the required engineering tolerance. Secondly we derived a new coupling condition for the mortar method which is able to replace an explicit resolution of an elastomer. This new transmission condition is obtained from a dimension reduction. We then compared the eigenvalues and eigenmodes computed within this approach to the conformingly discretized wall-slab example, the wall now being connected to the slab by means of an elastomer. The resulting lowest eight eigenvalues of the two models correspond within a tolerance of less than 1[ %]. This accuracy is sufficient for the application at hand. We finally demonstrate that the good results obtained by the newly developed mortar variant also extend to larger examples of engineering relevance.
The practical motivation of using the new mortar method was to greatly simplify both the engineering modeling effort and the meshing process by dispensing with the need for a conformal element coupling between construction components like slabs and walls. An interesting side effect, however, was that it was also possible to significantly reduce the overall computational workload. The conforming model of the engineering example resulted in 7578 hexahedral elements while only 2475 hexahedral elements were needed for the mortar model. This reduction is due to the facts that: a) a component-wise mesh generation naturally introduces the possibility to choose local mesh densities, b) necessary refinements in other building components do not need to be respected and, accordingly, do not spread across interfaces, and c) at the interfaces of orthogonally coupled, laminated structures it was possible to avoid unnecessary hexahedral elements naturally due to the relaxed topological constraints, and d) it is not required to resolve the geometrically thin elastomer layer.
Declarations
Acknowledgements
We would like to gratefully acknowledge the funds provided by the “Deutsche Forschungsgemeinschaft” under the contract/grant numbers: RA-624/21-1 and WO-671/13-1.
Authors’ Affiliations
References
- Bernardi C, Maday Y, Patera AT: A new non conforming approach to domain decomposition: the mortar element method. In Collége de France Seminar. Edited by: Brezis H, Lions J-L. Pitman, Paris, France, XI; 1994:13–51.Google Scholar
- Ben Belgacem F: The mortar finite element method with Lagrange multipliers. Numerische Mathematik 1999, 84: 173–197. 10.1007/s002110050468MATHMathSciNetView ArticleGoogle Scholar
- Wohlmuth BI: Discretization methods and iterative solvers based on domain decomposition. In Lecture Notes in Computational Science and Engineering. Springer, Berlin, New York; 2001.Google Scholar
- Hauret P, Tallec P: A discontinuous stabilized mortar method for general 3d elastic problems. Comput Methods Appl Mech Eng 2007,196(49–52):4881–4900. 10.1016/j.cma.2007.06.014MATHView ArticleGoogle Scholar
- Puso MA: A 3d mortar method for solid mechanics. Int J Numeric Methods Eng 2004, 59: 315–336. 10.1002/nme.865MATHView ArticleGoogle Scholar
- Wohlmuth BI, Popp A, Gee MW, Wall WA: An abstract framework for a priori estimates for contact problems in 3D with quadratic finite elements. Comput Mech 2012,49(6):735–747. 10.1007/s00466-012-0704-zMATHMathSciNetView ArticleGoogle Scholar
- Sitzmann S, Willner K, Wohlmuth BI: A dual lagrange method for contact problems with regularized contact conditions. Int J Numeric Methods Eng 2014, 99: 221–238. 10.1002/nme.4683MathSciNetView ArticleGoogle Scholar
- Casadei F, Gabellini E, Fotia G, Maggio F, Quarteroni A: A mortar spectral/finite element method for complex 2D and 3D elastodynamic problems. Comput Methods Appl Mech Eng 2002,191(45):5119–5148. 10.1016/S0045-7825(02)00294-3MathSciNetView ArticleGoogle Scholar
- Flemisch B, Wohlmuth B: Nonconforming methods for nonlinear elasticity problems. In Domain Decomposition Methods in Science and Engineering XVI. Lect. Notes Comput. Sci. Eng. Edited by: Widlund O, Keyes D. Springer, Berlin, Germany; 2007:65–76. 10.1007/978-3-540-34469-8_6View ArticleGoogle Scholar
- Hauret P, Tallec P: Two-scale Dirichlet-Neumann preconditioners for elastic problems with boundary refinements. Comput Methods Appl Mech Eng 2007,196(8):1574–1588. 10.1016/j.cma.2006.03.021MATHView ArticleGoogle Scholar
- Klöppel T, Popp A, Küttler U, Wall W: Fluid-structure interaction for non-conforming interfaces based on a dual mortar formulation. Comput Methods Appl Mech Eng 2011,200(45–46):3111–3126. 10.1016/j.cma.2011.06.006MATHView ArticleGoogle Scholar
- Peszynska M, Wheeler M, Yotov I: Mortar upscaling for multiphase flow in porous media. Comput Geoscience 2002,6(1):73–100. 10.1023/A:1016529113809MATHMathSciNetView ArticleGoogle Scholar
- Peszynska M: Mortar adaptivity in mixed methods for flow in porous media. Int J Numerical Anal Models 2005,2(3):241–282.MATHMathSciNetGoogle Scholar
- Triebenbacher S, Kaltenbacher M, Wohlmuth B, Flemisch B: Applications of the mortar finite element method in vibroacoustics and flow induced noise computations. Acta Acustica United Acustica 2010,96(3):536–553. 10.3813/AAA.918305View ArticleGoogle Scholar
- Flemisch B, Kaltenbacher M, Wohlmuth B: Elasto-acoustic and acoustic-acoustic coupling on non-matching grids. Int J Numeric Methods Eng 2006,67(13):1791–1810. 10.1002/nme.1669MATHMathSciNetView ArticleGoogle Scholar
- Buffa A, Perugia I, Warburton T: The mortar-discontinuous Galerkin method for the 2D Maxwell eigenproblem. J Sci Comput 2009, 40: 86–114. 10.1007/s10915-008-9238-0MATHMathSciNetView ArticleGoogle Scholar
- Lamichhane BP, Wohlmuth B: Mortar finite elements with dual Lagrange multipliers: some application. Lect Notes Comput Sci Eng 2005, 40: 319–326. 10.1007/3-540-26825-1_31MathSciNetView ArticleGoogle Scholar
- Seshaiyer P, Suri M: Uniform hp convergence results for the mortar finite element method. Math Comput 1999, 69: 521–546. 10.1090/S0025-5718-99-01083-2MathSciNetView ArticleGoogle Scholar
- Belgacem B, Seshaiyer P, Suri M: Optimal convergence rates of hp mortar finite element methods for second-order elliptic problems. ESAIM: Math Model Numerical Anal 2000, 34: 591–608. 10.1051/m2an:2000158MATHView ArticleGoogle Scholar
- Düster A, Rank E: The p-version of the Finite Element Method. In Encyclopedia of Computational Mechanics. Edited by: Stein E, de Borst R, Hughes TJR. John Wiley & Sons, Hoboken, New Jersey, USA; 2004:119–139.Google Scholar
- Rank E, Düster A, Nübel V, Preusch K, Bruhns OT: High order finite elements for shells. Comput Methods Appl Mech Eng 2005, 194: 2494–2512. 10.1016/j.cma.2004.07.042MATHView ArticleGoogle Scholar
- Rabold A (2010) Anwendung der finite element Methode auf die Trittschallberechnung. Dissertation, Chair for Computation in Engineering, Fakultät für Bauingenieur- und Vermessungswesen, Technische Universität München.Google Scholar
- Sorger C, Frischmann F, Kollmannsberger S, Rank E: TUM.GeoFrame: Automated high-order hexahedral mesh generation for shell-like structures. Eng Comput 2014,30(1):41–56. 10.1007/s00366-012-0284-8View ArticleGoogle Scholar
- Rüberg T, Martin Schanz M: Coupling finite and boundary element methods for static and dynamic elastic problems with non-conforming interfaces. Comput Methods Appl Mech Eng 2008, 198: 449–458. 10.1016/j.cma.2008.08.013MATHView ArticleGoogle Scholar
- Wang L, Hou S, Shi L: A numerical method for solving 3d elasticity equations with sharp-edged interfaces. Int J Partial Differential Equations 2013, 2013: 1–10. 10.1155/2013/476873View ArticleGoogle Scholar
- Hauret P, Ortiz M: Bv estimates for mortar methods in linear elasticity. Comput Methods Appl Mech Engrg 2006, 195: 4783–4793. 10.1016/j.cma.2005.09.021MATHMathSciNetView ArticleGoogle Scholar
- A finite element method for elasticity interface problems with locally modified triangulations Int J Numer Anal Model 2011,8(2):189–200.
- Finite element modeling of brick-mortar interface stresses Int J Civil Environ Eng 2012, 12: 48–67.
- Boström A, Bövik P, Olsson P: A comparison of exact first order and spring boundary conditions for scattering by thin layers. J Nondestructive Eval 1992, 11: 175–184. 10.1007/BF00566408View ArticleGoogle Scholar
- Bare DZ, Orlik J, Panasenko G: Asymptotic dimension reduction of a Robin-type elasticity boundary value problem in thin beams. Appl Anal 2014,93(6):1217–1238. 10.1080/00036811.2013.823481MATHMathSciNetView ArticleGoogle Scholar
- Wassouf Z (2010) The mortar method for the finite element method of high order. PhD thesis, Technische Universitä, t München. Wassouf Z (2010) The mortar method for the finite element method of high order. PhD thesis, Technische Universitä, t München.
- Gander MJ, Japhet C, Maday Y, Nataf F: A new cement to glue nonconforming grids with Robin interface conditions : the finite element case. Lect Notes Comput Sci Eng 2004, 40: 259–266. Springer, Berlin Heidelberg Springer, Berlin Heidelberg 10.1007/3-540-26825-1_24MathSciNetView ArticleGoogle Scholar
- Babuška I: Finite Element Analysis. John Wiley & Sons, Hoboken, New Jersey, USA; 1991.MATHGoogle Scholar
- Belgacem FB, Maday Y: The mortar element method for three dimensional finite elements. ESAIM: Mathematical Modelling and Numerical Analysis - Modé, lisation Mathématique et Analyse Numérique 1997,31(2):289–302.MATHGoogle Scholar
- Babuška I, Osborn J: Eigenvalue problems. Handbook Numerical Anal II 1991, 2: 642–787.Google Scholar
- Banks HT, Lybeck N: Modeling methodology for elastomer dynamics. Syst Control: Foundations Appl 1997, 22: 37–50.MathSciNetGoogle Scholar
- Castellani A, Kajon G, Panzeri P, Pezzoli P: Elastomeric materials used for vibration isolation of railway lines. J Eng Mech 1998, 124: 614–621. 10.1061/(ASCE)0733-9399(1998)124:6(614)View ArticleGoogle Scholar
- Pastor M, Binda M, Harcarik T (2012) Modal assurance criterion In: Procedia enginering MMaMS, 48, 543-548.Google Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.