XIGA: An eXtended IsoGeometric analysis approach for multi-material problems

Abstract

Multi-material problems often exhibit complex geometries along with physical responses presenting large spatial gradients or discontinuities. In these cases, providing high-quality body-fitted finite element analysis meshes and obtaining accurate solutions remain challenging. Immersed boundary techniques provide elegant solutions for such problems. Enrichment methods alleviate the need for generating conforming analysis grids by capturing discontinuities within mesh elements. Additionally, increased accuracy of physical responses and geometry description can be achieved with higher-order approximation bases. In particular, using B-splines has become popular with the development of IsoGeometric Analysis. In this work, an eXtended IsoGeometric Analysis (XIGA) approach is proposed for multi-material problems. The computational domain geometry is described implicitly by level set functions. A novel generalized Heaviside enrichment strategy is employed to accommodate an arbitrary number of materials without artificially stiffening the physical response. Higher-order B-spline functions are used for both geometry representation and analysis. Boundary and interface conditions are enforced weakly via Nitsche’s method, and a new face-oriented ghost stabilization methodology is used to mitigate numerical instabilities arising from small material integration subdomains. Two- and three-dimensional heat transfer and elasticity problems are solved to validate the approach. Numerical studies provide insight into the ability to handle multiple materials considering sharp-edged and curved interfaces, as well as the impact of higher-order bases and stabilization on the solution accuracy and conditioning.

XFEM level set-based topology optimization for turbulent conjugate heat transfer problems

“…where B k ( ) are B-spline basis functions and ( i ) k are the coefficients associated with the LSF i ( ) , similarly to Noël et al (2022). In contrast to advancing the LSFs in the optimization process through the solution of the Hamilton-Jacobi equation, see approaches by Sethian and Wiegmann (2000), Wang et al (2003), andAllaire et al (2004), the coefficients of the LSFs are expressed as explicit functions of the design (2)…”

“…In this paper, an XFEM approach is adopted, which allows for accurate and efficient resolution of multi-phase problems with evolving interfaces and boundaries. The approach is similar to the XIGA proposed in Noël et al (2022) but restricts the approximation of the state variable fields to first-order functions. The main ingredients of the XFEM approach are briefly recalled in this section.…”

“…In this paper, the enrichment strategy described in is extended to enrich each individual basis function separately based on the topology of the phase layout within the basis function support as described in Noël et al (2022). Considering a multi-phase problem, a state field ( ) is approximated as:…”

“…Standard quadrature is then performed on each triangular integration element and non-intersected background element. The reader is referred to Villanueva and Maute (2014) and Noël et al (2022) for further details on the subdivision strategy.…”

“…To discretize and solve conjugate heat transfer problems, an XFEM approach is adopted. This approach is similar to the XIGA approach presented in Noël et al (2022) but is restricted to first-order approximations for the state variable fields. Higher-order B-spline functions are used for the geometry representation as was done in Noël et al (2020).…”