A new space–time discretization for the Swift–Hohenberg equation that strictly respects the Lyapunov functional

Gómez, Héctor; Nogueira, Xesús

doi:10.1016/j.cnsns.2012.05.018

Cited by 55 publications

(45 citation statements)

References 52 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In these models, the phase-field or orderparameter varies smoothly (i.e., continuously) over these thin layers, with a key point being that the energy of these models needs to be dissipated as time progresses. The existence of a Lyapunov functional for these diffuseinterface problems implies strong energy stability [60], a property which can be lost if inadequate algorithms and/or spatio-temporal resolutions are used to solve the partial differential equation [24,48,55,60]. This work addresses the nonlinear stability issue for both conserved and non-conserved phase-field variables, and presents a simple process that relies on Taylor series to handle the nonlinear terms present in the partial differential equation.…”

Section: Energy Stability Of Phase-field Modelsmentioning

confidence: 99%

“…If inadequate discretizations are used to solve the partial differential equations involved, unphysical results where free energy increases can follow [24,54]. Considering equation (7), the jump in free energy 〚F〛 is defined as…”

Section: Phase-field Modelingmentioning

confidence: 99%

“…These equations were selected because their stable time integration has garnered considerable interest in recent years [23,24,52,54], and this work generalizes some of the numerical schemes that have been put forth in the context of the Cahn-Hilliard [27,36] and phase-field crystal equations [54]. Additionally, the discretization in space is done using isogeometric analysis [13], which allows to easily generate high-order and globally continuous basis functions.…”

Section: Phase-field Modelsmentioning

confidence: 99%

“…Numerical techniques that satisfy thermodynamic relations at the discrete level in phase-field models have been developed for both conserved [23,25] and non-conserved phase-field models [24,27]. This work generalizes some of these algorithms, and shows that the nonlinear terms present in the equations can be discretized in time using a Taylor series expansion to guarantee both second-order accuracy and strong energy stability.…”

Section: Introductionmentioning

confidence: 99%

“…The method has successfully been applied in its Galerkin version to solve the Cahn-Hilliard equation [21,23], the advective [39] and Navier-Stokes-CahnHilliard equations [19,56], the Swift-Hohenberg equation [24] and the phasefield crystal equation [25,54,57]. IGA possesses some advantages over standard finite element methods, which include being able to easily generate high-order, globally continuous basis functions as well as exact geometrical representations as the finite element space is refined.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

An energy-stable time-integrator for phase-field models

Vignal

Collier

Dalcín

et al. 2017

Computer Methods in Applied Mechanics and Engineering

View full text Add to dashboard Cite

We introduce a provably energy-stable time-integration method for general classes of phase-field models with polynomial potentials. We demonstrate how Taylor series expansions of the nonlinear terms present in the partial differential equations of these models can lead to expressions that guarantee energy-stability implicitly, which are second-order accurate in time. The spatial discretization relies on a mixed finite element formulation and isogeometric analysis. We also propose an adaptive time-stepping discretization that relies on a first-order backward approximation to give an error-estimator. This error estimator is accurate, robust, and does not require the computation of extra solutions to estimate the error. This methodology can be applied to any second-order accurate time-integration scheme. We present numerical examples in two and three spatial dimensions, which confirm the stability and robustness of the method. The implementation of the numerical schemes is done in PetIGA, a high-performance isogeometric analysis framework.

show abstract

Section: Energy Stability Of Phase-field Modelsmentioning

confidence: 99%

Section: Phase-field Modelingmentioning

confidence: 99%

Section: Phase-field Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An energy-stable time-integrator for phase-field models

Vignal

Collier

Dalcín

et al. 2017

Computer Methods in Applied Mechanics and Engineering

View full text Add to dashboard Cite

show abstract

Stability and convergence analysis for a new phase field crystal model with a nonlocal Lagrange multiplier

Lin¹,

Cao

Zhang

et al. 2020

Math Methods in App Sciences

View full text Add to dashboard Cite

In this work, an energy stable numerical scheme is proposed to solve the PFC model with a nonlocal Lagrange multiplier. The construction of the numerical scheme is based on invariant energy quadratization (IEQ) technique to transform a nonlinear system into a linear system, while the time variables are discretized by second‐order scheme. The stability term in the new scheme can balance the influence of nonlinear term. Moreover, we obtain the results of unconditional energy stability, uniqueness, and uniform boundedness of numerical solution, and the numerical scheme is convergent with order of scriptOfalse(δt2false). Several numerical tests are conducted to confirm the theoretical results.

show abstract

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

Casquero

Liu

Bona-Casas

et al. 2015

Int. J. Numer. Meth. Engng

Self Cite

View full text Add to dashboard Cite

Summary We present a hybrid variational‐collocation, immersed, and fully‐implicit formulation for fluid‐structure interaction (FSI) using unstructured T‐splines. In our immersed methodology, we define an Eulerian mesh on the whole computational domain and a Lagrangian mesh on the solid domain, which moves arbitrarily on top of the Eulerian mesh. Mathematically, the problem reduces to solving three equations, namely, the linear momentum balance, mass conservation, and a condition of kinematic compatibility between the Lagrangian displacement and the Eulerian velocity. We use a weighted residual approach for the linear momentum and mass conservation equations, but we discretize directly the strong form of the kinematic relation, deriving a hybrid variational‐collocation method. We use T‐splines for both the spatial discretization and the information transfer between the Eulerian mesh and the Lagrangian mesh. T‐splines offer us two main advantages against non‐uniform rational B‐splines: they can be locally refined and they are unstructured. The generalized‐α method is used for the time discretization. We validate our formulation with a common FSI benchmark problem achieving excellent agreement with the theoretical solution. An example involving a partially immersed solid is also solved. The numerical examples show how the use of T‐junctions and extraordinary nodes results in an accurate, efficient, and flexible method. Copyright © 2015 John Wiley & Sons, Ltd.

show abstract

A new space–time discretization for the Swift–Hohenberg equation that strictly respects the Lyapunov functional

Cited by 55 publications

References 52 publications

An energy-stable time-integrator for phase-field models

An energy-stable time-integrator for phase-field models

Stability and convergence analysis for a new phase field crystal model with a nonlocal Lagrange multiplier

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

Contact Info

Product

Resources

About