15 research outputs found
A Shallow Ritz Method for Elliptic Problems with Singular Sources
In this paper, a shallow Ritz-type neural network for solving elliptic
equations with delta function singular sources on an interface is developed.
There are three novel features in the present work; namely, (i) the delta
function singularity is naturally removed, (ii) level set function is
introduced as a feature input, (iii) it is completely shallow, comprising only
one hidden layer. We first introduce the energy functional of the problem and
then transform the contribution of singular sources to a regular surface
integral along the interface. In such a way, the delta function singularity can
be naturally removed without introducing a discrete one that is commonly used
in traditional regularization methods, such as the well-known immersed boundary
method. The original problem is then reformulated as a minimization problem. We
propose a shallow Ritz-type neural network with one hidden layer to approximate
the global minimizer of the energy functional. As a result, the network is
trained by minimizing the loss function that is a discrete version of the
energy. In addition, we include the level set function of the interface as a
feature input of the network and find that it significantly improves the
training efficiency and accuracy. We perform a series of numerical tests to
show the accuracy of the present method and its capability for problems in
irregular domains and higher dimensions
A parallel Voronoi-based approach for mesoscale simulations of cell aggregate electropermeabilization
We introduce a numerical framework that enables unprecedented direct
numerical studies of the electropermeabilization effects of a cell aggregate at
the meso-scale. Our simulations qualitatively replicate the shadowing effect
observed in experiments and reproduce the time evolution of the impedance of
the cell sample in agreement with the trends observed in experiments. This
approach sets the scene for performing homogenization studies for understanding
the effect of tissue environment on the efficiency of electropermeabilization.
We employ a forest of Octree grids along with a Voronoi mesh in a parallel
environment that exhibits excellent scalability. We exploit the electric
interactions between the cells through a nonlinear phenomenological model that
is generalized to account for the permeability of the cell membranes. We use
the Voronoi Interface Method (VIM) to accurately capture the sharp jump in the
electric potential on the cell boundaries. The case study simulation covers a
volume of with more than well-resolved cells with a
heterogeneous mix of morphologies that are randomly distributed throughout a
spheroid region.Comment: 23 pages, 19 figures, submitted to Journal of Computational Physic
An interpolation matched interface and boundary method for elliptic interface problems
AbstractAn interpolation matched interface and boundary (IMIB) method with second-order accuracy is developed for elliptic interface problems on Cartesian grids, based on original MIB method proposed by Zhou et al. [Y. Zhou, G. Wei, On the fictious-domain and interpolation formulations of the matched interface and boundary method, J. Comput. Phys. 219 (2006) 228–246]. Explicit and symmetric finite difference formulas at irregular grid points are derived by virtue of the level set function. The difference scheme using IMIB method is shown to satisfy the discrete maximum principle for a certain class of problems. Rigorous error analyses are given for the IMIB method applied to one-dimensional (1D) problems with piecewise constant coefficients and two-dimensional (2D) problems with singular sources. Comparison functions are constructed to obtain a sharp error bound for 1D approximate solutions. Furthermore, we compare the ghost fluid method (GFM), immersed interface method (IIM), MIB and IMIB methods for 1D problems. Finally, numerical examples are provided to show the efficiency and robustness of the proposed method
A cartesian grid finite volume method for the solution of the Poisson equation with variable coefficients and embedded interfaces
We present a finite volume method for the solution of the two-dimensional Poisson equation r· ((x)ru(x)) = f(x) with variable, discontinuous coefficients and solution
discontinuities on irregular domains. The method uses bilinear ansatz function on Cartesian grids for the solution u(x) resulting in a compact nine-point stencil.
The resulting linear problem has been solved with a standard multigrid solver. Singularities associated with vanishing partial volumes of intersected grid cells or the dual bilinear ansatz itself are removed by a two-step asymptotic approach. The method achieves second order of accuracy in the L1 and L2 norm
Solving elliptic problems with discontinuities on irregular domains – the Voronoi Interface Method.
We introduce a simple method, dubbed the Voronoi Interface Method, to solve Elliptic problems with discontinuities across the interface of irregular domains. This method produces a linear system that is symmetric positive definite with only its right-hand-side affected by the jump conditions. The solution and the solution's gradients are second-order accurate and first-order accurate, respectively, in the L∞L∞ norm, even in the case of large ratios in the diffusion coefficient. This approach is also applicable to arbitrary meshes. Additional degrees of freedom are placed close to the interface and a Voronoi partition centered at each of these points is used to discretize the equations in a finite volume approach. Both the locations of the additional degrees of freedom and their Voronoi discretizations are straightforward in two and three spatial dimensions
High-order numerical methods for 2D parabolic problems in single and composite domains
In this work, we discuss and compare three methods for the numerical
approximation of constant- and variable-coefficient diffusion equations in both
single and composite domains with possible discontinuity in the solution/flux
at interfaces, considering (i) the Cut Finite Element Method; (ii) the
Difference Potentials Method; and (iii) the summation-by-parts Finite
Difference Method. First we give a brief introduction for each of the three
methods. Next, we propose benchmark problems, and consider numerical tests-with
respect to accuracy and convergence-for linear parabolic problems on a single
domain, and continue with similar tests for linear parabolic problems on a
composite domain (with the interface defined either explicitly or implicitly).
Lastly, a comparative discussion of the methods and numerical results will be
given.Comment: 45 pages, 12 figures, in revision for Journal of Scientific Computin
An efficient mass-preserving interface-correction level set/ghost fluid method for droplet suspensions under depletion forces
Aiming for the simulation of colloidal droplets in microfluidic devices, we present here a numerical method for two-fluid systems subject to surface tension and depletion forces among the suspended droplets. The algorithm is based on an efficient solver for the incompressible two-phase Navier–Stokes equations, and uses a mass-conserving level set method to capture the fluid interface. The four novel ingredients proposed here are, firstly, an interface-correction level set (ICLS) method; global mass conservation is achieved by performing an additional advection near the interface, with a correction velocity obtained by locally solving an algebraic equation, which is easy to implement in both 2D and 3D. Secondly, we report a second-order accurate geometric estimation of the curvature at the interface and, thirdly, the combination of the ghost fluid method with the fast pressurecorrection approach enabling an accurate and fast computation even for large density contrasts. Finally, we derive a hydrodynamic model for the interaction forces induced by depletion of surfactant micelles and combine it with a multiple level set approach to study short-range interactions among droplets in the presence of attracting forces