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2Boundary integration
2Scaled boundary finite element method
10102 Applied Mathematics
10802 Computation Theory and Mathematics
109 Engineering
10905 Civil Engineering
10913 Mechanical Engineering
1Decomposable
1Minkowski sums
1Polygonal finite element method
1Polytopes
1Smoothed finite element method
1Virtual element method
1Wachspress shape functions

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Some indecomposable polyhedra

**Authors:**Yost, David**Date:**2007**Type:**Text , Journal article**Relation:**Optimization Vol. 56, no. 5-6 (2007), p. 715-724**Full Text:****Reviewed:****Description:**We complete the classification, in terms of decomposability, of all combinatorial types of polytopes with 14 or fewer edges. Recall that a polytope P is said to be decomposable if it is equal to a Minkowski sum [image omitted] of two polytopes Q and R which are not similar to P. Our main contribution here is to consider the 42 types of polyhedra with 8 faces and 8 vertices. It turns out that 34 of these are always indecomposable, and 5 are always decomposable. The remaining 3 are ambiguous, i.e. each of them has both decomposable and indecomposable geometric realizations.**Description:**C1**Description:**2003004904

**Authors:**Yost, David**Date:**2007**Type:**Text , Journal article**Relation:**Optimization Vol. 56, no. 5-6 (2007), p. 715-724**Full Text:****Reviewed:****Description:**We complete the classification, in terms of decomposability, of all combinatorial types of polytopes with 14 or fewer edges. Recall that a polytope P is said to be decomposable if it is equal to a Minkowski sum [image omitted] of two polytopes Q and R which are not similar to P. Our main contribution here is to consider the 42 types of polyhedra with 8 faces and 8 vertices. It turns out that 34 of these are always indecomposable, and 5 are always decomposable. The remaining 3 are ambiguous, i.e. each of them has both decomposable and indecomposable geometric realizations.**Description:**C1**Description:**2003004904

A scaled boundary finite element formulation over arbitrary faceted star convex polyhedra

- Natarajan, Sundararajan, Ooi, Ean Tat, Saputra, Albert, Song, Chongmin

**Authors:**Natarajan, Sundararajan , Ooi, Ean Tat , Saputra, Albert , Song, Chongmin**Date:**2017**Type:**Text , Journal article**Relation:**Engineering Analysis with Boundary Elements Vol. 80, no. (2017), p. 218-229**Full Text:**false**Reviewed:****Description:**In this paper, a displacement based finite element framework for general three-dimensional convex polyhedra is presented. The method is based on a semi-analytical framework, the scaled boundary finite element method. The method relies on the definition of a scaling center from which the entire boundary is visible. The salient feature of the method is that the discretizations are restricted to the surfaces of the polyhedron, thus reducing the dimensionality of the problem by one. Hence, an explicit form of the shape functions inside the polyhedron is not required. Conforming shape functions defined over arbitrary polygon, such as the Wachpress interpolants are used over each surface of the polyhedron. Analytical integration is employed within the polyhedron. The proposed method passes patch test to machine precision. The convergence and the accuracy properties of the method is discussed by solving few benchmark problems in linear elasticity. © 2017 Elsevier Ltd

- Natarajan, Sundararajan, Bordas, Stéphane, Ooi, Ean Tat

**Authors:**Natarajan, Sundararajan , Bordas, Stéphane , Ooi, Ean Tat**Date:**2015**Type:**Text , Journal article**Relation:**International Journal for Numerical Methods in Engineering Vol. 104, no. 13 (2015), p. 1173-1199**Full Text:**false**Reviewed:****Description:**We show both theoretically and numerically a connection between the smoothed finite element method (SFEM) and the virtual element method and use this approach to derive stable, cheap and optimally convergent polyhedral FEM. We show that the stiffness matrix computed with one subcell SFEM is identical to the consistency term of the virtual element method, irrespective of the topology of the element, as long as the shape functions vary linearly on the boundary. Using this connection, we propose a new stable approach to strain smoothing for polygonal/polyhedral elements where, instead of using sub-triangulations, we are able to use one single polygonal/polyhedral subcell for each element while maintaining stability. For a similar number of degrees of freedom, the proposed approach is more accurate than the conventional SFEM with triangular subcells. The time to compute the stiffness matrix scales with the O(dofs)1.1 in case of the conventional polygonal FEM, while it scales as O(dofs)0.7 in the proposed approach. The accuracy and the convergence properties of the SFEM are studied with a few benchmark problems in 2D and 3D linear elasticity. © 2015 John Wiley & Sons, Ltd.

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