Functional Entropy Variables: A New Methodology for Deriving Thermodynamically Consistent Algorithms for Complex Fluids, with Particular Reference to the Isothermal Navier-Stokes-Korteweg Equations

Abstract

Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. We propose a new methodology for the numerical solution of the isothermal Navier-Stokes-Korteweg equations. Our methodology is based on a semi-discrete Galerkin method invoking functional entropy variables, a generalization of classical entropy variables, and a new time integration scheme. We show that the resulting fully discrete scheme is unconditionally stable-in-energy, second-order time-accurate, and mass-conservative. We utilize isogeometric analysis for spatial discretization and verify the aforementioned properties by adopting the method of manufactured solutions and comparing coarse mesh solutions with overkill solutions. Various problems are simulated to show the capability of the method. Our methodology provides a means of constructing unconditionally stable numerical schemes for nonlinear non-convex hyperbolic systems of conservation laws. AbstractWe propose a new methodology for the numerical solution of the isothermal Navier-StokesKorteweg equations. Our methodology is based on a semi-discrete Galerkin method invoking functional entropy variables, a generalization of classical entropy variables, and a new time integration scheme. We show that the resulting fully discrete scheme is unconditionally stable-in-energy, second-order time-accurate, and mass-conservative. We utilize isogeometric analysis for spatial discretization and verify the aforementioned properties by adopting the method of manufactured solutions and comparing coarse mesh solutions with overkill solutions. Various problems are simulated to show the capability of the method. Our methodology provides a means of constructing unconditionally stable numerical schemes for nonlinear non-convex hyperbolic systems of conservation laws.

A hybrid variational-collocation immersed method for fluid-structure interaction using unstructured T-splines

“…NURBS‐based IGA is completely widespread nowadays. An important aftereffect of using NURBS as a basis in analysis is that we can take advantage of their natural inter‐element smoothness, which has positive consequences in several application areas, such as, body‐fitted fluid‐structure interaction (FSI) , immersed FSI , fluid mechanics , phase‐field models , biomechanics , structural mechanics , shape memory alloys , shell modeling , and contact problems , among others.…”

An energy‐stable mixed formulation for isogeometric analysis of incompressible hyperelastodynamics

“…However, we do not favor this type of “energy” estimates because the pressure‐squared term does not carry physical meanings. To remedy this issue, an entropy variable can be introduced by leveraging the convexity of the volumetric energy, and a physically relevant entropy stability is expected . This is beyond the scope of this work and remains an area of future research.…”

Parametric mesh regularization for interpolatory shape design and isogeometric analysis over a mesh of arbitrary topology

“…Dating back to 2006, Bazilevs et al [3] did a mathematical study of Isogeometric Analysis based on NURBS. After that, many researchers have taken NURBS as the basis for IGA applications such as fluid mechanics [4][5][6][7][8][9][10] structural mechanics [11][12][13][14][15][16][17], shape optimization [18][19][20][21][22] and so on. Other splines are also proposed for IGA, such as T-splines [23][24][25][26][27][28][29], locally-refined splines [30], generalized B-splines [31,32], PHT-splines [33][34][35], RHT-splines [36] and so on.…”