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A Novel Supermesh Method for Computing Solutions to the Multi-material Stefan Problem with Complex Deforming Interfaces and Microstructure

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Abstract

A novel supermesh method that was presented for computing solutions to the multimaterial heat equation in complex stationary geometries with microstructure, is now applied for computing solutions to the Stefan problem involving complex deforming geometries with microstructure. The supermesh is established by combining the underlying (fixed) structured rectangular grid with the (deforming) piecewise linear interfaces reconstructed by the multi-material moment-of-fluid method. The temperature diffusion equation with Dirichlet boundary conditions at interfaces is solved by the linear exact multi-material finite volume method upon the supermesh. The interface propagation equation is discretized using the unsplit cell-integrated semi-Lagrangian method. The level set method is also coupled during this process in order to assist in the initialization of the (transient) provisional velocity field. The resulting method is validated on both canonical and challenging benchmark tests. Algorithm convergence results based on grid refinement are reported. It is found that the new method approximates solutions to the Stefan problem efficiently, compared to traditional approaches, due to the localized finite volume approximation stencil derived from the underlying supermesh. The new deforming boundary supermesh approach enables one to compute many kinds of complex deforming boundary problems with the efficiency properties of a body fitted mesh and the robustness of a “cut-cell” (a.k.a. “embedded boundary” or “immersed”) method.

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Data Availability Statement

Data will be made available on reasonable request.

Abbreviations

e :

Internal energy

\(\rho \) :

Density

\(C_v\) :

Specific heat at constant volume

T :

Time

\(\kappa \) :

Thermal conductivity

\(\theta \) :

Temperature

\(\theta _{sat}\) :

Saturation temperature

\(\varvec{x},\varvec{y}\) :

Physical coordinate

\(\varGamma \) :

Interface

\(\phi \) :

Signed distance function

\({\varvec{n}}\) :

Normal

V :

Velocity

L :

Latent heat

b :

Intercept

\(E,\epsilon \) :

Error

M :

Number of materials

\(F,F_{1},F_{2},\ldots ,F_{M}\) :

Material volume fraction

\(\varvec{x}_{1},\varvec{x}_{2},\ldots ,\varvec{x}_{M}\) :

Material centroid

\(\varOmega , \varOmega _{1},\varOmega _{2},\ldots ,\varOmega _{M}\) :

Material domain(s)

\(m,m1,m2,\ldots \) :

Material number

q :

Heat flux

\(\omega \) :

Weights

A :

Area

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Funding

This material is based upon work supported by National Aeronautics and Space Administration under the Grant Nos. NNX16AQ65G S003, 80NSSC20K0352.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Joint College of Engineering Florida State University-Florida A&M University, Tallahassee, FL, 32310, USA

    Yang Liu & Kourosh Shoele

  2. Department of Mathematics, Florida State University, Tallahassee, FL, 32306, USA

    Mark Sussman & M. Yousuff Hussaini

  3. Department of Mechanical Engineering, University of Louisville, Louisville, KY, 40292, USA

    Yongsheng Lian

  4. Mechanical & Aerospace Engineering, MathWorks Inc., 3 Apple Hill Drive, Natick, MA, 01760, USA

    Mehdi Vahab

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  1. Yang Liu
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Liu, Y., Sussman, M., Lian, Y. et al. A Novel Supermesh Method for Computing Solutions to the Multi-material Stefan Problem with Complex Deforming Interfaces and Microstructure. J Sci Comput 91, 19 (2022). https://doi.org/10.1007/s10915-022-01783-1

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