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Multi-criteria gridshell optimization
Published in Sigrid Adriaenssens, Philippe Block, Diederik Veenendaal, Chris Williams, Shell Structures for Architecture, 2014
The first step is to discretize the given surface (initially represented with NURBS, see Figure 15.3) as a triangulated surface with triangular facets. Practically, one way to do this is to use a proprietary FE surface meshing tool (although it is never directly used for any FE analysis). This discretization facilitates the use of discrete differential geometry to synthesize and manipulate grid geometry on the surface.
An autonomous intelligent framework for optimal orientation detection in 3D printing
Published in International Journal of Computer Integrated Manufacturing, 2023
Mohammad Reza Rezaei, Mahmoud Houshmand, Omid Fatahi Valilai
To solve the second issue, the general approach in the discrete differential geometry is to assign curvature measures to vertices based on the behavior of the shape around that vertex (Crane et al. 2013). To accomplish this many different heuristic approaches may be applied. One of these approaches named discrete mean curvature measure from (Cohen-Steiner and Marie Morvan 2003) is applied in the pictures of the first columns of Figure 5 This method assigns a curvature value to each vertex of the shape with respect to a neighborhood radius and based on the normal vectors of the planes which intersect in that vertex and is applied in Trimesh© library of Python©. The signs of resulting curvature values are used to tag each vertex as being either convex or concave (positive or negative concavity respectively). The solution to the first issue can be found after the definition of the curvature for each vertex, by determining the vertices which are identified as being concave, and by creating loops from neighboring concave vertices in the adjacency graph. Figure 5 represents some shapes created by combination of simpler shapes, the calculated curvature in their vertices and the resulting loops from the adjacency graph. As can be seen in the Figure, cutting the shape using the created loops from concave vertices, many shapes can be decomposed into simpler shapes.
A discrete nonlocal damage mechanics approach
Published in Mechanics of Advanced Materials and Structures, 2022
Arun R. Srinivasa, J. N. Reddy, Nam Phan
Cohesive zone elements have numerous issues with loss of stability and spurious waves in dynamic problems, as well as convergence issues. They need to be implemented carefully. On the other hand, the Extended FEM method (XFEM) [10], while being able to resolve individual cracks through elements, requires extensive modifications of conventional codes and is not suited for the micro-cracking phenomena that occurs in composites where the problem is not dominated by a single crack but a distribution of microcracks. Furthermore, our recent comparisons have indicated that the XFEM approach is not suitable for modeling multiple intersecting cracks, and in many cases the approach fails to converge (see, e.g., Thamburaja et al. [8, 13]).the GraFEA approach: The starting point of our approach is based on the observation of the underlying unity behind all physics based models that “irrespective of the way in which the underlying smooth continuum based physics is obtained, all computational models reduce to the specification of a finite number of (discrete) degrees of freedom located at specific points (nodes) in the domain and an algebraic relationship between the nodal degrees of freedom and suitably defined dual variables that (the duality) is determined by physical principles. The approach described here, called GraFEA (for Graph Based Finite Element Analysis; see [14, 15]) combines this insight with discrete differential geometry to provide a sound basis for carrying out the method.
Numerical convergence of discrete exterior calculus on arbitrary surface meshes
Published in International Journal for Computational Methods in Engineering Science and Mechanics, 2018
Mamdouh S. Mohamed, Anil N. Hirani, Ravi Samtaney
Avoiding the circumcentric dual on non-Delaunay triangulations became the common practice in discrete differential geometry and exterior calculus implementations. For example, when defining discrete differential geometry operators, such as normal vector and curvatures, Meyer et al. [23] reverted locally to the barycentric dual mesh whenever a non-Delaunay triangle pair exists. Moreover, DEC discretizations of the incompressible Navier–Stokes equations [3], [7] employed the circumcentric dual mesh and consequently considered only Delaunay triangulations for the application of the developed schemes. Furthermore, Mullen et al. [24] pointed out that for a Delaunay/Voronoi triangulation, failure to keep the circumcenter inside its triangle/tetrahedron can lead to numerical degeneracy. Hirani et al. [25] showed that for a certain choice of sign convention for circumcentric dual objects, the discrete Hodge star assembled based on such elementary dual pieces is positive definite if the primal mesh is Delaunay. Delaunay meshes are therefore sufficient for obtaining positive entries in the diagonal Hodge matrices. In that paper, no mathematical claim was made regarding the solution accuracy in case the Delaunay condition was violated. However, a numerical experiment in Hirani et al. [25] showed the solution of a scalar Poisson equation on a non-Delaunay triangulation (using the circumcentric dual) to be incorrect. Later on, an error in the code was discovered [26]. After correcting the code, the solution on both Delaunay and non-Delaunay meshes appeared to be similar and correct.