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Condition Number Variation. Condition numbers for the stiffness matrices on the unit disc assembled using 500 different shifts of the background grid (h = 0.13).
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We apply the recently developed least squares stabilized symmetric Nitsche method for enforcement of Dirichlet boundary conditions to the finite cell method. The least squares stabilized Nitsche method in combination with finite cell stabilization leads to a symmetric positive definite stiffness matrix and relies only on elementwise stabilization,...
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... also note that diagonal scaling seems to work very well as a preconditioner for the method. In Figure 8 we get a view of how the condition numbers vary over 500 different cut situations for the disc on a fixed mesh size, and we see the least-squares stabilization seems to have a very positive effect for larger values of τ . Again, we see the effectiveness of diagonal scaling. ...
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Citations
... For this, so-called ghost penalty stabilization is used, introduced in [3] and generalized to higher-order in [20]. Still, there are other viable options for stabilizing a Nitsche-type method, for instance, discrete extension [5], least-squares stabilization [8,18], or cell merging [1,12]. ...
... . Whereas the bounds for δ above are devised for optimal order approximation properties, this bound is typically suboptimal for optimal order scaling of the stiffness matrix condition number, cf. [18] where similar bounds are computed to scale an inconsistent type of stabilization. An optimal bound for the condition number would likely involve a somewhat less aggressive scaling of δ, and the region of regularizationω δ using such a bound is reasonably the region where the method will run into numerical issues. ...
We consider elliptic problems in multipatch IGA where the patch parameterizations may be singular. To deal with this issue, we develop a robust weak formulation that allows trimmed (cut) elements, enforces interface and Dirichlet conditions weakly, and does not depend on specially constructed approximation spaces. Our technique for dealing with the singular maps is based on the regularization of the Riemannian metric tensor, and we detail how to implement this robustly. We investigate the method's behavior when applied to a square-to-cusp parameterization that allows us to vary the singular behavior's aggressiveness. We propose a scaling of the regularization parameter to obtain optimal order approximation. Our numerical experiments indicate that the method is robust also for quite aggressive singular parameterizations.
... One advantage of the FCM is that it can be utilized on differ- Fig. 1 Mesh generation for a heterogeneous material using FEM and FCM ent geometric models, such as implicit geometry description (level-set functions) [45,49,95], B-rep models (STL) [27,33,74], voxel models (CT scans) [31,40,50,96], or constructive solid geometry (CSG) [78,91,92], to name a few. Recently, the FCM was further developed to solve additional challenges arising in cut finite cells, such as the numerical integration of the stiffness matrix and load vector [3,32,44], the imposition of boundary conditions [16,54,62], and the solution of linear systems of equations [19][20][21]50]. For smooth problems, the FCM exhibits high convergence rates and hence, can compete with the p-FEM [28,71]. ...
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... The numerical challenges faced in these methods are typically caused by the fact that elements are intersected by the boundaries (see Fig. 1c), leading to important simulation aspects, such as the 1. Integration of discontinuous element matrices (FCM: [5,[14][15][16][17], XFEM: [18][19][20][21], CutFEM: [22]), 2. Stabilization schemes for ill-conditioned problems (FCM: [23][24][25], XFEM: [13,26,27], CutFEM: [28,29]), 3. Implementation of weak boundary conditions on immersed boundaries (FCM: [30], XFEM: [13], CutFEM: [31]), 4. Accurate capturing of weakly or strongly discontinuous displacement fields (FCM: [32], XFEM: [33,34], Cut-FEM: [6]), 5. Mass lumping and time integration schemes for transient problems (SCM: [10,35]), to just name a few. The list given above demonstrates the multitude of important topics related to immersed boundary methods, which are entire research areas on their own. ...
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... Instead, similarly to immersed methods such as the Finite Cell Method (FCM) [33,34], or the Isogeometric B-Rep Analysis (IBRA) [35] essential boundary conditions are usually processed in a weak form. Common approaches in other FEM and immersed FEM methods are the penalty approach ( [36] for FEM, e.g., [37] for FCM, e.g., [35,38] for IBRA, e.g., [39] for MPM), the Nitsche method (e.g., [40] for FCM, e.g., [41] for IGA), Mortar-based methods (e.g., [42] for FEM, e.g., [43] for FCM, [44] for IGA), or the Lagrange multiplier approach ( [45] for FEM, e.g., [38,41] for IGA/IBRA). All methods have their advantages in different numerical scenarios. ...
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... • A least squares stabilization term can be applied when the employed function space is (at least) C 1 -continuous. In [185] and [186] it is demonstrated that a stable and coercive formulation can also be achieved by applying a least squares finite element term in the vicinity of the Dirichlet boundary. Because in H 2 (Ω) the normal gradient on the boundary can be controlled by volumetric terms, coercivity is even achieved in the full space instead of only in the discrete space. ...
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This thesis presents innovative methodological developments for a
seamless computer-aided design-integrated simulation and structural
assessment of masonry in a numerical multiphysics environment.
A numerically classified experimental program is processed to obtain a
quantitative classification of masonry, for which a novel computational
constitutive law is developed and presented. Furthermore, finite
element-based physics are enhanced to cope with the properties of
computer-aided design descriptions towards the demands of masonry
structures. With the presented avenues increased possibilities arise
in the structural assessment of buildings, including post-damage
load-carrying behaviors and limit stress states for various impact
scenarios. This is demonstrated in a selection of relevant benchmark
problems. With the presented choice of small-scale tests, an eventual
solution scheme towards the predictability by simulation of historic
and new buildings shall be introduced.
The development of the computer-aided design-integrated methods is
presented in a manner that reaches beyond the needs of masonry and
shall enlighten the general applicability of the proposed approaches.
The established technologies are introduced along with realized code
developments, specifically addressing a unified implementation for
a generic integration within various finite element software environments.
