Approximation of the Laplace and Stokes operators with Dirichlet boundary conditions through volume penalization: a spectral viewpoint

Abstract

Abstract. We report the results of a detailed study of the spectral properties of Laplace and Stokes operators, modified with a volume penalization term designed to approximate Dirichlet conditions in the limit when a penalization parameter, η, tends to zero. The eigenvalues and eigenfunctions are determined either analytically or numerically as functions of η, both in the continuous case and after applying Fourier or finite difference discretization schemes. For fixed η, we find that only the part of the spectrum corresponding to eigenvalues λ η −1 approaches Dirichlet boundary conditions, while the remainder of the spectrum is made of uncontrolled, spurious wall modes. The penalization error for the controlled eigenfunctions is estimated as a function of η and λ. Surprisingly, in the Stokes case, we show that the eigenfunctions approximately satisfy, with a precision O(η), Navier slip boundary conditions with slip length equal to √ η. Moreover, for a given discretization, we show that there exists a value of η, corresponding to a balance between penalization and discretization errors, below which no further gain in precision is achieved. These results shed light on the behavior of volume penalization schemes when solving the Navier-Stokes equations, outline the limitations of the method, and give indications on how to choose the penalization parameter in practical cases.

Volume penalization for inhomogeneous Neumann boundary conditions modeling scalar flux in complicated geometry

“…There is no intermediate region at low k, as it is the case for the VP representation of homogeneous Neumann boundary conditions [5]. The absence of the region is in contrast to what is observed for the VP representation of the Dirichlet boundary condition [19], and is thus attributed to the absence of any boundary layer due to the penalization of Neumann boundary conditions.…”

“…(1), (19) and (20), and the penalized equation (22) are obtained by using the second order finite-differences and interpolation in Eqs. (17) and (18).…”

Directional change of fluid particles in two-dimensional turbulence and of football players

“…In the nonperiodic case the circular boundaries are imposed by a volume penalization method [13,14]. The penalization technique introduces a numerical parameter η which is chosen proportional to x 2 (with x the grid size), to minimize the error [15]. Thereby the modeling error of the no-slip conditions is of the same order as the discretization error.…”

Adaptive wavelet simulation of weakly compressible flow in a channel with a suddenly expanded section

“…Thus, it is important to choose a suitable value of η for a given grid size, taking into account a balance between the CPU cost and the accuracy of the solutions. We also refer to the discussion in [20]. For the cases of η = 10 −3 and 10 −4 , ∆t is determined by the constraints imposed by η, while for the case of η = 10 −2 , ∆t is determined by the diffusive constraint.…”