Quadrature-free immersed isogeometric analysis

Abstract

This paper presents a novel method for solving partial differential equations on three-dimensional CAD geometries by means of immersed isogeometric discretizations that do not require quadrature schemes. It relies on a newly developed technique for the evaluation of polynomial integrals over spline boundary representations that is exclusively based on analytical computations. First, through a consistent polynomial approximation step, the finite element operators of the Galerkin method are transformed into integrals involving only polynomial integrands. Then, by successive applications of the divergence theorem, those integrals over B-Reps are transformed into the first surface and then line integrals with polynomials integrands. Eventually, these line integrals are evaluated analytically with machine precision accuracy. The performance of the proposed method is demonstrated by means of numerical experiments in the context of 2D and 3D elliptic problems, retrieving optimal error convergence order in all cases. Finally, the methodology is illustrated for 3D CAD models with an industrial level of complexity.

Image‐based isogeometric twins of lattices with virtual image correlation for varying cross‐section beams

“…Given their versatility with respect to the geometric complexity of the specimen, these methods have been applied in many fields, such as for the analysis of trabecular bones, 30,31 foams, 32,33 composite materials, 34,35 porous media, 36 and AM lattices 18,24 . However, it is worth noting that (i) they only lead to an implicit representation of the geometry and (ii) their numerical complexity may be important, especially the numerical integration of cut elements remains challenging from an efficiency point of view 37,38 . On the other hand, template‐fitting methods build a parameterized spline‐based template geometry representative of the essential features of the specimen, and are accompanied by a control point fitting procedure to accurately match the scan data.…”

“…18,24 However, it is worth noting that (i) they only lead to an implicit representation of the geometry and (ii) their numerical complexity may be important, especially the numerical integration of cut elements remains challenging from an efficiency point of view. 37,38 On the other hand, template-fitting methods build a parameterized spline-based template geometry representative of the essential features of the specimen, and are accompanied by a control point fitting procedure to accurately match the scan data. Hence, these techniques require starting from a description whose topology is compatible with that of the scanned object but, in turn, they have the advantage of providing an explicit compact, IGA-suitable CAD description.…”

Unfittted finite element methods for explicit boundary representations

“…Such intersection is illustrated in Figure 2.8. 2 We can readily use B D and B N instead, to compute B cut D and B cut N . In this intersection, since the half-spaces related to faces(K) are processed, they are eliminated from H (see Algorithm 2).…”

“…The resulting convex polyhedron-surface components are intersected using Algorithm 2 and added to the sub-mesh for K that represent its interior part in line 11. 2 We note that one could also extract the surface mesh as the surface of the clipped polytopes obtained after intersecting again half-spaces in faces(S) in line 12. This is the reason why we use K •• in line 7 (to process surface faces aligned with background cells faces only once), closed spaces in lines 7-8 of Algorithm 4 (not to lose any surface face after splitting with wall half-spaces) and intersection against open half-spaces related to the surfaces in line 11 (to discard zero volume components after this decomposition and count surface faces on walls only once).…”

“…Similarly to unfitted FE methods, immersed IGA [2,3,119] eliminates the need for unstructured meshes by utilizing the intersection of a background mesh. These methods utilize integration techniques for complex domains, including dimension reduction of integrands [40,57], i.e., integrating over lines and surfaces.…”