686 research outputs found
Entropy-Production-Rate-Preserving Algorithms for Thermodynamically Consistent Nonisothermal Models of Incompressible Binary Fluids
We derive a thermodynamically consistent, non-isothermal, hydrodynamic model
for incompressible binary fluids following the generalized Onsager principle
and Boussinesq approximation. This model preserves not only the volume of each
fluid phase but also the positive entropy production rate under
thermodynamically consistent boundary conditions. Guided by the thermodynamical
consistency of the model, a set of second order structure-preserving numerical
algorithms are devised to solve the governing partial differential equations
along with consistent boundary conditions in the model, which preserve the
entropy production rate as well as the volume of each fluid phase at the
discrete level. Several numerical simulations are carried out using an
efficient adaptive time-stepping strategy based on one of the
structure-preserving schemes to simulate the Rayleigh-B\'{e}nard convection in
the binary fluid and interfacial dynamics between two immiscible fluids under
competing effects of the temperature gradient, gravity, and interfacial forces.
Roll cell patterns and thermally induced mixing of binary fluids are observed
in a rectangular region with insulated lateral boundaries and vertical ones
with imposed temperature difference. Long time simulations of interfacial
dynamics are performed demonstrating robust results of new structure-preserving
schemes
Arbitrarily High-Order Unconditionally Energy Stable Schemes for Thermodynamically Consistent Gradient Flow Models
We present a systematic approach to developing arbitrarily high-order, unconditionally energy stable numerical schemes for thermodynamically consistent gradient flow models that satisfy energy dissipation laws. Utilizing the energy quadratization method, we formulate the gradient flow model into an equivalent form with a corresponding quadratic free energy functional. Based on the equivalent form with a quadratic energy, we propose two classes of energy stable numerical approximations. In the first approach, we use a prediction-correction strategy to improve the accuracy of linear numerical schemes. In the second approach, we adopt the Gaussian collocation method to discretize the equivalent form with a quadratic energy, arriving at an arbitrarily high-order scheme for gradient flow models. Schemes derived using both approaches are proved rigorously to be unconditionally energy stable. The proposed schemes are then implemented in four gradient flow models numerically to demonstrate their accuracy and effectiveness. Detailed numerical comparisons among these schemes are carried out as well. These numerical strategies are rather general so that they can be readily generalized to solve any thermodynamically consistent PDE models
Thermodynamically consistent space-time discretization of non-isothermal mechanical systems in the framework of GENERIC
The present work addresses the design of structure-preserving numerical methods that emanate from the general equation for non-equilibrium reversible-irreversible coupling (GENERIC) formalism. Novel energy-momentum (EM) consistent time-stepping schemes in the realm of molecular dynamics are proposed. Moreover, the GENERIC-based structure-preserving numerical methods are extended to the context of large-strain thermoelasticity and thermo-viscoelasticity
A thermodynamically consistent model of magneto-elastic materials under diffusion at large strains and its analysis
The theory of elastic magnets is formulated under possible diffusion and heat
flow governed by Fick's and Fourier's laws in the deformed (Eulerian)
configuration, respectively. The concepts of nonlocal nonsimple materials and
viscous Cahn-Hilliard equations are used. The formulation of the problem uses
Lagrangian (reference) configuration while the transport processes are pulled
back. Except the static problem, the demagnetizing energy is ignored and only
local non-selfpenetration is considered. The analysis as far as existence of
weak solutions of the (thermo)dynamical problem is performed by a careful
regularization and approximation by a Galerkin method, suggesting also a
numerical strategy. Either ignoring or combining particular aspects, the model
has numerous applications as ferro-to-paramagnetic transformation in elastic
ferromagnets, diffusion of solvents in polymers possibly accompanied by
magnetic effects (magnetic gels), or metal-hydride phase transformation in some
intermetalics under diffusion of hydrogen accompanied possibly by magnetic
effects (and in particular ferro-to-antiferromagnetic phase transformation),
all in the full thermodynamical context under large strains
Thermodynamically consistent space-time discretization of non-isothermal mechanical systems in the framework of GENERIC
The present work addresses the design of structure-preserving numerical methods that emanate from the general equation for non-equilibrium reversible-irreversible coupling (GENERIC) formalism. Novel energy-momentum (EM) consistent time-stepping schemes in the realm of molecular dynamics are proposed. Moreover, the GENERIC-based structure-preserving numerical methods are extended to the context of large-strain thermoelasticity and thermo-viscoelasticity
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