The local tangential lifting method for moving interface problems on surfaces with applications

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Abstract

In this paper, a new numerical computational frame is presented for solving moving interface problems modeled by parabolic PDEs on static and evolving surfaces. The surface PDEs can have Dirac delta source distributions and discontinuous coefficients. One application is for thermally driven moving interfaces on surfaces such as Stefan problems and dendritic solidification phenomena on solid surfaces. One novelty of the new method is the local tangential lifting method to construct discrete delta functions on surfaces. The idea of the local tangential lifting method is to transform a local surface problem to a local two dimensional one on the tangent planes of surfaces at some selected surface nodes. Moreover, a surface version of the front tracking method is developed to track moving interfaces on surfaces. Strategies have been developed for computing geodesic curvatures of interfaces on surfaces. Various numerical examples are presented to demonstrate the accuracy of the new methods. It is also interesting to see the comparison of the dendritic solidification processes in two dimensional spaces and on surfaces.

Introduction

Many physical phenomena can be modeled by partial differential equations (PDEs) on static or evolving surfaces that are not necessarily planes. As a matter of fact, the earth's surface is close to a sphere as we know. There is a large collection of literatures on PDEs on surfaces, or surface PDEs. In this paper, we propose a numerical study for moving interface problems on static and evolving surfaces, that is, the PDE is defined on a static or evolving surface, on which a co-dimension one interface is moving with time.

In the past decade and nowadays, various numerical methods have been developed for solving PDEs on surfaces. Among them, the surface finite element method [8], [9], [41] approximates a surface using a triangular mesh. Then the PDEs can be solved by the linear finite element space on the triangular mesh. The closest point method [26], [27], [33] uses an embedding approach to solve surface PDEs in a three dimensional tube with extensions of surface quantities and function values to a neighborhood of the surfaces. Then the problem can be solved on a three dimensional domain using finite difference or other numerical methods. The local tangential lifting (LTL) method presented in [4], [5], [39] is the one that is closest to our method in this paper. Given a surface problem and a partition, in a neighborhood of a node, the LTL method transforms the surface problem locally to a tangent plane of the surface by extending function values to the tangent planes. The local discretization results on tangent planes are regarded as those of original problem on the surface. Thus, the total discretization is composed of those discretizations on the tangent planes and can be solved accordingly. In the original version of LTL method, the local stencils, see [39] for example, on tangent planes are obtained by lifting the triangular meshes of surfaces. In fact, an alternative way is to construct local stencils on two dimensional planes locally. With this idea, our LTL method can be viewed as a local mesh method that has the merits of a meshfree method, see [44]. Therefore, in this paper, the LTL method is employed to discretize surface diffusion terms in PDEs, which avoids a global surface triangulation and maintains the accuracy of the algorithm. Some stability analysis is also presented for the proposed method.

As a practical application, we also apply our method to classical Stefan or solidification problems on surfaces in which solutions represent temperature. There are plenty of papers on two- and three-dimensional thermally driven Stefan and solidification problems, see [3], [12], [18], [23], [25], [36], [38], [50] for an incomplete list. However, few research can be found in the literature on numerical method for such interface problems on surfaces. In this paper, we develop a surface version of front tracking method for the thermally driven moving interface so that we can control the discretization errors from the interface discretization and evolving. There are several advantages of the developed front tracking method. The moving interface in a surface problem using the front tracking method is explicitly tracked on a number of selected markers. Therefore, the locations of the interfaces can be obtained at all times; the interface markers can be directly used to compute the geodesic curvature. With this approach, interface conditions such as the Stefan condition can be satisfied explicitly. Note also that it is natural to combine the LTL method with the front tracking method. This is because the LTL method has the merits of a meshfree method. We can move the interfaces directly on the original implicit surfaces, instead of on a surface triangular mesh, which is more difficult in implementation. Moreover, another challenge is how to deal with singular source distributions represented by Dirac delta functions and the discontinuity in heat equations on surfaces. In this paper, we apply the immersed boundary (IB) method [30], [31] and develop a non-trivial extension of IB method on surfaces using the idea of LTL method.

For the space-time discretization of PDEs on evolving surfaces, the Lagrangian particle method is a popular numerical approach [2], [8], [35] that can compute the material derivative from moving particles. Thus, the movement of evolving surface can be tracked explicitly. Based on the LTL method, we apply the Lagrangian particle approach to solve the PDEs with singular source distributions on evolving surfaces. Moreover, combining with the proposed front tracking method, we develop a computation framework for the simulation of classical Stefan/solidification problems on evolving surfaces.

We believe that other methods, such as the combination of surface finite element methods with level set method [29] or volume of fluid methods [17], [32], would work as well if details can be workout with advantages and limitations. Other related works include the level set method on surfaces [24], [26], [40], phase field methods for moving interfaces on surfaces [6], [7], [19], [21], [42], [43], the simulation of the coupled bulk-surface problems [1], [10], [15], [13], and surfactant computations [20], [28], [45], [46], [47] etc.

The layout of the paper is organized as follows. In Section 2, we introduce the model problems concerned in this paper. In Section 3, the LTL method for surface parabolic PDEs and the discrete delta functions are presented. In Section 4, we focus on implementation details of the front tracking method for moving interfaces on static surfaces. In Section 5, the computational frame for moving interface problems on evolving surfaces is explained. Numerical examples are presented in Section 6 including numerical simulations of dendritic solidification phenomena on surfaces. We conclude and point out some further research in the last section.

Section snippets

The model problems

In this section, we introduce some moving interface problems modeled by parabolic PDEs on static and evolving surfaces.

The local tangential lifting method

In this section, we explain the local tangential lifting method for the discretizations of the diffusion operator and the Dirac delta function on surfaces. We assume that we have obtained a sufficiently dense and uniformly distributed node set {xi}i=1N on Γ. This can be done by a marching method proposed in [16].

The front tracking method for moving interfaces on surfaces

In this section, we focus on the implementation of the front tracking method for the moving interface problems on surfaces. The front tracking method used in this paper consists of interface updating, interface smoothing, geodesic curvature computation, and interface velocity computation.

The computation frame of moving interface problems on evolving surfaces

In this section, we consider the numerical simulation of moving solid-liquid interfaces on evolving surfaces. There are two major steps. One is how to solve the surface PDE, the other one is about an iterative method to compute the interface velocity.

Numerical experiments

In this section, we present several numerical examples to validate our proposed numerical methods.

Conclusions and further studies

In this work, a new numerical computational frame is proposed to solve parabolic PDEs on static and evolving surfaces with singular source distributions defined along the moving interfaces. Novelties of this paper include the LTL method to construct a discrete surface diffusion operator, a non-trivial extension of the IB method on surfaces, the front tracking method for moving interfaces on static and evolving surfaces, and an iterative method for computing interface velocity in the dendritic

CRediT authorship contribution statement

Xufeng Xiao: Conceptualization, Methodology, Software, Visualization. Xinlong Feng: Formal analysis, Investigation, Methodology, Project administration, Supervision, Validation. Zhilin Li: Supervision, Validation, Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The first author would like to thank the Department of Mathematics, North Carolina State University for hosting Xufeng Xiao's visit between 2019 and 2020 supported by the funding of China Scholarship Council. Part of the work in this paper was performed during the visit. The first author would like also to thank the doctoral research fellowships of Xinjiang University's and Tianchi program. The authors would like to thank the editor and referees for their valuable comments and suggestions which

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