Highly accurate surface and volume integration on implicit domains by means of moment‐fitting

Abstract

SUMMARY We introduce a new method for the numerical integration over curved surfaces and volumes defined by a level set function. The method is based on the solution of a small linear system based on a simplified variant of the moment‐fitting equations. Numerical experiments suggest that the accuracy of the resulting quadrature rules exceeds the accuracy of traditional methods by orders of magnitude. Using moments up to an order of p, the measured experimental orders of convergence exceed hp. Consequently, their construction is very efficient because only coarse computational grids are required. The conceptual simplicity allows for the application on very general grid types, which is demonstrated by numerical experiments on quadrilateral, triangular and hexahedral grids. Copyright © 2013 John Wiley & Sons, Ltd.

An unstructured immersed finite element method for nonlinear solid mechanics

“…The cut elements are general polytopes for which quadrature rules are not easily obtained. There are many techniques that address this problem, such as moment-fitting [40], surface-only integration [41], and adaptive decomposition of the integration region [14,42]. But since we have a tessellation in simplex shapes already available, we use composite Gauß type quadrature rules, see e.g., [11].…”

High-Order Quadrature Methods for Implicitly Defined Surfaces and Volumes in Hyperrectangles

“…In that work, it was assumed that the surface integral integrand, g, could be smoothly extended off the surface Γ and that the normal vector field induced by φ, i.e., ∇φ/|∇φ|, was smooth throughout U ; with these assumptions, a moment-fitting method was used to choose the best representative forũ coming from a finite-dimensional space of divergence-free vector-valued functions via a least-squares problem. In particular, for p quadrature nodes per dimension (i.e., p d in total for each U ), approximate convergence rates of order p + 1 were demonstrated [17]. We note, however, that techniques which use integration by parts may not directly work if the boundary term is empty, i.e., Ω ∩ ∂U = ∅.…”

“…The resulting integrals are themselves integrals over implicitly defined domains, and so the procedure can be repeated in one fewer dimensions, leading to a recursive scheme on the number of spatial dimensions. Müller, Kummer, and Oberlack [17] used this technique to construct high-order quadrature schemes for quadrilateral, triangular, and hexahedral elements. In that work, it was assumed that the surface integral integrand, g, could be smoothly extended off the surface Γ and that the normal vector field induced by φ, i.e., ∇φ/|∇φ|, was smooth throughout U ; with these assumptions, a moment-fitting method was used to choose the best representative forũ coming from a finite-dimensional space of divergence-free vector-valued functions via a least-squares problem.…”

A parameter-free variational coupling approach for trimmed isogeometric thin shells

“…Isogeometric shells have been successfully applied for large-deformation analysis [21], in conjunction with various nonlinear material models [22,23], and in contact and fluid-structure interaction problems [24][25][26][27]. On the embedded domain side, the importance of geometrically faithful quadrature of trimmed elements and corresponding techniques have been discussed in a series of recent papers [28,27,[29][30][31][32][33][34][35]. For the weak enforcement of boundary and interface conditions at trimming curves and surfaces, variational methods such as Lagrange multiplier [36][37][38] or Nitsche techniques [39][40][41][42][43][44] have been successfully developed.…”