Journal of Computational Physics 233 (2013) 241–261 Contents lists available at SciVerse ScienceDirect Journal of Computational Physics journal homepage: www.elsevier.com/locate/jcp A level set approach for diffusion and Stefan-type problems with Robin boundary conditions on quadtree/octree adaptive Cartesian grids ⇑ Joseph Papac a, , Asdis Helgadottir b, Christian Ratsch a, Frederic Gibou b,c a Mathematics Department, University of California, Los Angeles, CA 91405, United States b Mechanical Engineering Department, University of California, Santa Barbara, CA 93106, United States c Computer Science Department, University of California, Santa Barbara, CA 93106, United States article info abstract Article history: We present a numerical method for simulating diffusion dominated phenomena on irreg- Received 22 November 2011 ular domains and free moving boundaries with Robin boundary conditions on quadtree/ Received in revised form 24 August 2012 octree adaptive meshes. In particular, we use a hybrid finite-difference and finite-volume Accepted 25 August 2012 framework that combines the level-set finite difference discretization of Min and Gibou Available online 13 September 2012 (2007) [13] with the treatment of Robin boundary conditions of Papac et al. (2010) [19] on uniform grids. We present numerical results in two and three spatial dimensions on Keywords: the diffusion equation and on a Stefan-type problem. In addition, we present an application Level set method of this method to the case of the simulation of the Ehrlich–Schwoebel barrier in the context Epitaxial growth Diffusion of epitaxial growth. Stefan problem Published by Elsevier Inc. Sharp interface Robin boundary condition 1. Introduction Diffusion and Stefan-type problems with Robin boundary conditions are of practical significance in a variety of fields. For example, these equations arise in heat transfer applications involving internal conduction and convection [11]; tissue imag- ing with near-infrared tomography [21]; and continuum models of epitaxial growth involving the Ehrlich–Schwoebel barrier [28] on step edges. The mathematical formulation of a diffusion equation is as follows: consider a domain X ¼ Xþ [ XÀ with boundary @X, illustrated in Fig. 1, where the solution u satisfying the diffusion equation, ut ¼ DMu þ g; ð1Þ where D is the diffusion coefficient and g represents source terms, is to be solved in XÀ. At the boundary of XÀ, which we denote by C, a Robin boundary condition is imposed: ru Á n þ au ¼ f ; x 2 C; ð2Þ where n is the outward normal to XÀ. The mathematical formulation of the Stefan problem is a free boundary problem where the velocity of the free boundary is given by: ⇑ Corresponding author. Tel.: +1 3102063570. E-mail address: [email protected] (J. Papac). 0021-9991/$ - see front matter Published by Elsevier Inc. http://dx.doi.org/10.1016/j.jcp.2012.08.038 242 J. Papac et al. / Journal of Computational Physics 233 (2013) 241–261 Fig. 1. A schematic of the level-set representation of the domain. V ¼½DruC; ð3Þ where ½C denotes a jump across the interface. The phases on each side of the free boundary satisfy the diffusion equation (1) with boundary condition at the free boundary given by Eq. (2). The solution itself and the coefficients associated to each phase can be discontinuous across the interface. A level-set approach to solving these equations on arbitrarily-shaped domains with uniform Cartesian grids was proposed in [19], where the treatment of the Robin boundary condition follows a finite volume approach [22,15]. The approach was introduced in two spatial dimensions and results in second-order accurate solutions for diffusion problems and first-order accurate solutions for Stefan-type problems. In this work, the numerical approach is extended to adaptive Cartesian grids in two and three spatial dimensions and we present an application to the simulation of the Ehrlich–Schwoebel barrier in the context of the island dynamics model of Caflisch et al. [4]. Some of the beneficial attributes of this approach are that it is straightforward to implement, computationally efficient, and geometrically robust so that complex interface topology and motion of sharp interfaces are handled implicitly. 2. Numerical approach 2.1. Level set method In the case of Stefan-type problems, the evolution is driven by the physical interaction at the interface between phases. For this reason, it is desirable to have a sharp interface method, where the location of the interface can be precisely defined. In this work, we utilize the level-set method [18] for implicitly representing the moving interface. One well-documented limitation of the level-set method is its propensity for mass loss when the grid is too coarse, since the method is not intrin- sically conservative. The discretization of these equations on non-graded adaptive grids allows for very fine resolution near the interface that significantly reduces mass loss while producing a computationally efficient method. A brief description of the level-set method follows and we refer the reader to [17,25] for more details. Referring to the domain depicted in Fig. 1, we describe XÀ by the set of points, x, such that /ðxÞ < 0. Likewise, we describe Xþ by the set of points such that /ðxÞ > 0. The interface C is implicitly defined by the zero level set, /ðxÞ¼0. The evolution of the interface is then given by the evolution of the level-set function, /, and obeys: /t þ V Á r/ ¼ 0; ð4Þ where V is an externally generated velocity field. The outward unit normal to the interface, n, and the interface mean curvature, j, are calculated from the level-set func- tion according to, r/ n ¼ and j ¼ r Á n; jr/j respectively. 2.2. Structure of the adaptive Cartesian grid We utilize non-graded Cartesian grids, those for which the size ratio between adjacent cells is unconstrained. The com- putational domain, X, is encompassed entirely within a root cell, a cube in three spatial dimensions or a square in two spatial dimensions, with grid nodes located at the vertices of the cell. The root cell is then recursively divided into equally-sized J. Papac et al. / Journal of Computational Physics 233 (2013) 241–261 243 sub-cells until the desired level of refinement is achieved, i.e. a parent cell is split into four child cells in two spatial dimen- sions and eight child cells in three spatial dimensions. This approach provides great flexibility in defining a suitable grid. In the simplest configuration, the domain can be divided into a standard uniform grid. The primary benefit, however, is the abil- ity to locally refine as necessary to capture small-scale physical details, or to coarsen the grid in areas where the solution is smooth to reduce the computational cost. While the criterion for refining the size of cells may be tailored specifically to the application at hand, in this work we have chosen a simple algorithm for generating the grid as proposed in [13,6], which is based on an estimate of the distance of each cell to the interface. More precisely, a cell is divided if the following condition is satisfied: min j/ðvÞj < Lip  DiagðCÞ; ð5Þ v2vertices where /ðvÞ is the value of the level-set function at the vertices, v, of the grid cell, C. Lip is the Lipschitz constant, and DiagðCÞ is the length of the diagonal of the cell. In our examples, we have chosen a value of Lip ¼ 1:1 since our reinitialization algo- rithm produces level-set functions that are approximate signed distance functions. Additionally, we define two grid param- eters, max level and min level that control the sizes of the largest and the smallest leaf cells in the domain: defining max level in a domain ½0; 12 corresponds to setting Dx ¼ Dy ¼ 1=2max level. Lastly, we impose a narrow band of uniform cells near the inter- face as a requirement for the discretization procedure. We do this by applying the criterion in (5) with the / values increased by ÆDx, Æ2Dx, etc., where Dx is the size of the smallest cell’s width. The node-based Cartesian grid described above is well-suited for tree-based data structures; octree in three spatial dimensions and quadtree in two spatial dimensions. These data structures ease implementation by providing an efficient and straightforward way to access and store the data [23,24]. 2.3. Approach for diffusion problems Consider a diffusion equation (1) with boundary condition given by (2). We will utilize a discretization based on the Crank–Nicolson scheme in time. The spatial discretizations are given in the following sections. 2.3.1. Discretization in three spatial dimensions We utilize a discretization in time based on the Crank–Nicolson scheme. When discretizing Eq. (1) in space and forming the associated linear system, we treat the grid nodes adjacent to the interface differently from the other nodes. First, consider an octree grid node that is not adjacent to the interface. The most general case contains at most one two- dimensional T-junction and one three-dimensional T-junction, as depicted in Fig. 2. The value of the ghost node due to the two-dimensional T-junction is found by compensating the error of linear interpolation by the derivative in the transverse direction, as described in [13,5], and is given by, u7s8 þ u8s7 s7s8 u6 À u0 u3 À u0 u2 ¼ À þ : s7 þ s8 s3 þ s6 s6 s3 Fig. 2. Octree grid showing one two-dimensional T-junction node, u2, and one three-dimensional T-junction node, u4. 244 J. Papac et al. / Journal of Computational Physics 233 (2013) 241–261 The value of the ghost node due to the three-dimensional T-junction is obtained similarly and is given by the following expression, s11s12u11 þ s12s9u12 þ s10s11u9 þ s10s9u10 s10s12 u3 À u0 u6 À u0 s9s11 u2 À u0 u5 À u0 u4 ¼ À þ À þ : ðs10 þ s12Þðs11 þ s9Þ s3 þ s6 s3 s6 s2 þ s5 s2 s5 Once all of the necessary ghost nodes are defined, the first and second-order derivatives are calculated with the following formulas: 0 u4 À u0 s1 u0 À u1 s4 Dx u0 ¼ Á þ Á s4 s1 þ s4 s1 s1 þ s4 and 0 u4 À u0 2 u0 À u1 2 Dxxu0 ¼ Á À Á ; s4 s1 þ s4 s1 s1 þ s4 and are used for constructing the linear system associated with the Crank–Nicolson time discretization of Eq.