A diffuse modeling approach for embedded interfaces in linear elasticity

Abstract

In this contribution, we present a diffuse modeling approach to embed material interfaces into nonconforming meshes with a focus on linear elasticity. For this purpose, a regularized indicator function is employed that describes the distribution of the different materials by a scalar value. The material in the resulting diffuse interface region is redefined in terms of this indicator function and recomputed by a homogenization of the adjacent material parameters. The applied homogenization method fulfills the kinematic compatibility across the interface and the static equilibrium at the interface. In addition, an hℓ‐adaptive refinement strategy based on truncated hierarchical B‐spline is applied to provide an appropriate and efficient approximation of the diffuse interface region. We justify mathematically and demonstrate numerically that the applied approach leads to optimal convergence rates in the far field for one‐dimensional problems. A two‐dimensional example illustrates that the application of the hℓ‐adaptive refinement strategy allows for a clear reduction of the error in the near and far field and a good resolution of the local stress and strain fields at the interface. The use of a higher continuous B‐spline basis leads to efficient computations due to the higher continuity of the diffuse interface model.

A Selection of Benchmark Problems in Solid Mechanics and Applied Mathematics

“…Especially discretization methods that avoid geometry conforming meshes, as presented e.g. in [33][34][35] will find interesting challenges in the presented two-dimensional benchmarks. Some extensions towards three dimensions can be found in [36].…”

Adaptive isogeometric discretizations for diffuse modeling of discontinuities

“…1. Six deformation states for three-dimensional and three deformation states for two-dimensional problems have to be considered to compute the components of the resulting effective material tensor C(c) [11]. The indicator function c and 1 − c are interpreted as the dimension inȳ direction of the individual phases.…”

“…[11] 3 Phase-field modeling of interface failure However, the total error can be clearly reduced under local hrefinement and an improved convergence rate is obtained compared to the embedded sharp interface under local h-refinement.…”

“…The indicator function c and 1 − c are interpreted as the dimension inȳ direction of the individual phases. Six deformation states for three-dimensional and three deformation states for two-dimensional problems have to be considered to compute the components of the resulting effective material tensor C(c) [11]. To take the orientation of the material interface into account, the material tensorC(c) is transformed from the reference coordinate system to the material coordinate system with basis {µ Γ , ν Γ , ξ Γ }.…”

“…In case of weak discontinuities we apply an hadaptive refinement strategy, where the interface width and the element size h are reduced simultaneously. This allows for an improved convergence to the sharp interface solution and a good resolution of the local stress and strain fields [11].…”