Quality Measures for Curvilinear Finite Elements

Abstract

We present a method for computing robust shape quality measures defined for finite elements of any order and any type, including curved pyramids. The measures are heuristically defined as the minimum of the pointwise quality of curved elements. Three pointwise qualities are considered: the ICN that is related to the conditioning of the stiffness matrix for straight-sided simplicial elements, the scaled Jacobian that is defined for quadrangles and hexahedra, and a new shape quality that is defined for triangles and tetrahedra. Based on previous work presented by

Combining High-Order Metric Interpolation and Geometry Implicitization for Curved r-Adaption

“…Following, we apply the r-conjunction operation for each hyperplane H to obtain the convex-hull representation γ CH(Γ) , see Equation (15). Finally, we obtain its normalized version γCH(Γ) by applying Equation (13).…”

“…In Algorithm 1, we describe how to obtain the implicit representation γ Λ of a model Λ. In Line 2, we compute for each Bézier patch Γ i of Λ its implicit function γ Γ i and we normalize it, see Equations ( 12) and (13). Then, in Line 3, we consider the convex hull property of the Bézier control points for trimming [29].…”

“…We first obtain an implicit representation of the convex hull γ CH(Γ i ) by applying a pair-wise r-conjunction to the hyperplane functions, see Equation (16). Then, we compute its normalized representation γCH(Γ i ) , see Equation (13). In Line 4, we trim the Bézier patch representation γΓ i in terms of γCH(Γ i ) , see Equation (14).…”

“…Herein, we consider the gradient and Hessian of the normalized representation γ, presented in Equation (13). As before, we denote by ∇f * ∇g the matrix with coefficients ∂ j f ∂ k g for j, k = 1, ..., d. In addition, we consider the symmetric term ∇f ⊗ ∇g := ∇f * ∇g + ∇g * ∇f given by the matrix with coefficients We observe that they require the first, second, and third derivatives of γ.…”

“…To efficiently match curved solution features, many practitioners have recently started to exploit curved high-order meshes. These meshes can be stretched and aligned in a pointwise varying fashion through anisotropic procedures [12], geodesic approaches for curved edges [13,14], shock-tracking methods [15,16,17], and deformation analogies [18,19]. Alternatively, the curved r-adaption can be driven, as for straight-edged elements [4,5], by distortion measures.…”