Immersed Boundary Smooth Extension (IBSE): A high-order method for solving incompressible flows in arbitrary smooth domains

Abstract

The Immersed Boundary method is a simple, efficient, and robust numerical scheme for solving PDE in general domains, yet for fluid problems it only achieves first-order spatial accuracy near embedded boundaries for the velocity field and fails to converge pointwise for elements of the stress tensor. In a previous work we introduced the Immersed Boundary Smooth Extension (IBSE) method, a variation of the IB method that achieves high-order accuracy for elliptic PDE by smoothly extending the unknown solution of the PDE from a given smooth domain to a larger computational domain, enabling the use of simple Cartesian-grid discretizations. In this work, we extend the IBSE method to allow for the imposition of a divergence constraint, and demonstrate high-order convergence for the Stokes and incompressible Navier-Stokes equations: up to third-order pointwise convergence for the velocity field, and second-order pointwise convergence for all elements of the stress tensor. The method is flexible to the underlying discretization: we demonstrate solutions produced using both a Fourier spectral discretization and a standard second-order finite-difference discretization.

Projection-based reduced order models for a cut finite element method in parametrized domains

“…As an alternative, one could flip the role of extension and transportation, first transporting snapshots and then extending the transported snapshots; we omit this case, as it only affects the harmonic extension case with negligible impact on the overall results and discussion. 4 In particular, we transport the elliptical geometries (corresponding to the snapshots for various parameters µ during the offline stage) to circular ones with fixed radius. Indeed, the transformation x → µ 1 x + µ 3 , y → µ 2 y + µ 4 guarantees that the ellipse µ 2 2 (x − µ 3 ) 2 + µ 2 1 (y − µ 4 ) 2 − µ 2 1 µ 2 2 R 2 = 0 is mapped into the circle x 2 + y 2 − R 2 = 0.…”

“…Recent improvements go under the names of Ghost-Cell finite difference methods, Cut-Cell finite volume approach, Immersed Interface, Ghost Fluid, Volume Penalty methods, for which we refer to the review paper [1] and references within. In particular, for what concerns incompressible flows in arbitrary smooth domains, the Immersed Boundary Smooth Extension method has shown high-order convergence for the incompressible Navier-Stokes equations [4].More in detail, extended mesh finite element methods using cut elements are examined in [5,6] for stationary Stokes flow systems, as well as for Navier-Stokes. An analysis for high Reynolds numbers, independent of the local Reynolds, has been carried out in [7,8,9].…”

Coarse graining the dynamics of immersed and driven fiber assemblies

“…The unit-tangent n and T are discretized at half-integer nodes, and the position X and force density F are computed at whole-integer nodes. The Stokes equations may be solved in a variety of ways depending on the domain; for general domains we use the Immersed Boundary Smooth Extension method with C 2 extensions, described in [36], which provides third-order accuracy of the velocity field in L ∞ . The primary challenges in this formulation arise from communication between frames (described in Section III 2), and the efficient time-stepping of the semi-discretized system (described in Section III 3).…”

Very high‐order accurate finite volume scheme for the convection‐diffusion equation with general boundary conditions on arbitrary curved boundaries

“…The reference point corresponds to the edge midpoint, m ∶ = m i j , and the stencil,  ∶=  i , gathers s cells in the vicinity of the edge. The solution of the associated unconstrained least-squares problem (9) provides vector̃i that minimizes cost functional (8), and the associated unconstrained polynomial reconstruction is denoted as̃i (x) =̃T i p d (x − m i ).…”

“…There are very few methods capable of handling curved domains with polygonal meshes and most of them are limited to the first-or second-order convergence of accuracy. Recently, an extension of the immersed boundary method to the fourth-order convergence of accuracy has been proposed in the framework of the Fourier spectral method, 8,9 which is able of handling arbitrary smooth curved domains.The ROD method was initially developed for the steady-state two-dimensional convection-diffusion problem with Dirichlet boundary conditions, and subsequent improvements are reported in this article, namely, the introduction of Neumann and Robin boundary conditions, which represents an important advance for real context applications; and the development of a generic framework to compute polynomial reconstructions based on the least-squares method, which allows to handle general constraints and improves upon the algorithm.The remaining sections of this article are organized as follows. Section 2 presents the model, the mesh, and the basic assumptions and notations.…”