@article{wan_li_2012, title={SOME NEW FINITE DIFFERENCE METHODS FOR HELMHOLTZ EQUATIONS ON IRREGULAR DOMAINS OR WITH INTERFACES}, volume={17}, ISSN={["1531-3492"]}, DOI={10.3934/dcdsb.2012.17.1155}, abstractNote={Solving a Helmholtz equation Δu + λu = f efficiently is a challenge for many applications. For example, the core part of many efficient solvers for the incompressible Navier-Stokes equations is to solve one or several Helmholtz equations. In this paper, two new finite difference methods are proposed for solving Helmholtz equations on irregular domains, or with interfaces. For Helmholtz equations on irregular domains, the accuracy of the numerical solution obtained using the existing augmented immersed interface method (AIIM) may deteriorate when the magnitude of λ is large. In our new method, we use a level set function to extend the source term and the PDE to a larger domain before we apply the AIIM. For Helmholtz equations with interfaces, a new maximum principle preserving finite difference method is developed. The new method still uses the standard five-point stencil with modifications of the finite difference scheme at irregular grid points. The resulting coefficient matrix of the linear system of finite difference equations satisfies the sign property of the discrete maximum principle and can be solved efficiently using a multigrid solver. The finite difference method is also extended to handle temporal discretized equations where the solution coefficient λ is inversely proportional to the mesh size.}, number={4}, journal={DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B}, author={Wan, Xiaohai and Li, Zhilin}, year={2012}, month={Jun}, pages={1155–1174} }
 @article{wan_li_lubkin_2008, title={Mechanics of mesenchymal contribution to clefting force in branching morphogenesis}, volume={7}, ISSN={["1617-7959"]}, url={http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcAuth=ORCID&SrcApp=OrcidOrg&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=WOS:000258612200008&KeyUID=WOS:000258612200008}, DOI={10.1007/s10237-007-0105-y}, abstractNote={Branching morphogenesis is ubiquitous and may involve several different mechanisms. Glandular morphogenesis is affected by growth, cell rearrangements, changes in the basal lamina, changes in the stromal ECM, changes in cell-cell and cell-ECM adhesions, mesenchymal contractility, and possibly other mechanisms. We have developed a 3D model of the mechanics of clefting, focusing in this paper solely on the potential role of mesenchyme-generated traction forces. The tissue mechanics are assumed to be those of fluids, and the hypothesized traction forces are modeled as advected by the deformations which they generate. We find that mesenchymal traction forces are sufficient to generate a cleft of the correct size and morphology, in the correct time frame. We find that viscosity of the tissues affects the time course of morphogenesis, and also affects the resulting form of the organ. Morphology is also strongly dependent on the initial distribution of contractility. We suggest an in vitro method of examining the role of mesenchyme in branching morphogenesis.}, number={5}, journal={BIOMECHANICS AND MODELING IN MECHANOBIOLOGY}, author={Wan, Xiaohai and Li, Zhilin and Lubkin, Sharon R.}, year={2008}, month={Oct}, pages={417–426} }
 @article{li_wan_ito_lubkin_2006, title={An augmented approach for the pressure boundary condition in a Stokes flow}, volume={1}, number={5}, journal={Communications in Computational Physics}, author={Li, Z. L. and Wan, X. H. and Ito, K. and Lubkin, S. R.}, year={2006}, pages={874–885} }
 @article{lubkin_wan_2006, title={Optimizing detection of tissue anisotropy by fluorescence recovery after photobleaching}, volume={68}, ISSN={["1522-9602"]}, url={http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcAuth=ORCID&SrcApp=OrcidOrg&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=WOS:000241796800003&KeyUID=WOS:000241796800003}, DOI={10.1007/s11538-006-9074-z}, abstractNote={Fluorescence recovery after photobleaching (FRAP) has been widely used to measure fluid flow and diffusion in gels and tissues. It has not been widely used in detection of tissue anisotropy. This may be due to a lack of applicable theory, or due to inherent limitations of the method. We discuss theoretical aspects of the relationship between anisotropy of tissue structure and anisotropy of diffusion coefficients, with special regard to the size of the tracer molecule used. We derive a semi-mechanistic formula relating the fiber volume fraction and ratio of fiber and tracer molecule diameters to the expected anisotropy of the diffusion coefficients. This formula and others are tested on simulated random walks through random simulated and natural media. We determine bounds on the applicability of FRAP for detection of tissue anisotropy, and suggest minimum tracer sizes for detection of anisotropy in tissues of different composition (fiber volume fraction and fiber diameter). We find that it will be easier to detect anisotropy in monodisperse materials than in polydisperse materials. To detect mild anisotropy in a tissue, such as cartilage, which has a low fiber fraction would require a tracer molecule so large that it would be difficult to deliver to the tissue. We conclude that FRAP can be used to detect tissue anisotropy when the tracer molecule is sufficiently large relative to the fiber diameter, volume fraction, and degree of polydispersivity, and when the anisotropy is sufficiently pronounced.}, number={8}, journal={BULLETIN OF MATHEMATICAL BIOLOGY}, author={Lubkin, S. R. and Wan, X.}, year={2006}, month={Nov}, pages={1873–1891} }