203 research outputs found
Characterizing the stabilization size for semi-implicit Fourier-spectral method to phase field equations
Recent results in the literature provide computational evidence that
stabilized semi-implicit time-stepping method can efficiently simulate phase
field problems involving fourth-order nonlinear dif- fusion, with typical
examples like the Cahn-Hilliard equation and the thin film type equation. The
up-to-date theoretical explanation of the numerical stability relies on the
assumption that the deriva- tive of the nonlinear potential function satisfies
a Lipschitz type condition, which in a rigorous sense, implies the boundedness
of the numerical solution. In this work we remove the Lipschitz assumption on
the nonlinearity and prove unconditional energy stability for the stabilized
semi-implicit time-stepping methods. It is shown that the size of stabilization
term depends on the initial energy and the perturba- tion parameter but is
independent of the time step. The corresponding error analysis is also
established under minimal nonlinearity and regularity assumptions
On the well posedness of static boundary value problem within the linear dilatational strain gradient elasticity
In this paper, it is proven an existence and uniqueness theorem for weak solutions of the equilibrium problem for linear isotropic dilatational strain gradient elasticity. Considered elastic bodies have as deformation energy the classical one due to Lamé but augmented with an additive term that depends on the norm of the gradient of dilatation: only one extra second gradient elastic coefficient is introduced. The studied class of solids is therefore related to Korteweg or Cahn–Hilliard fluids. The postulated energy naturally induces the space in which the aforementioned well-posedness result can be formulated. In this energy space, the introduced norm does involve the linear combination of some specific higher-order derivatives only: it is, in fact, a particular example of anisotropic Sobolev space. It is also proven that aforementioned weak solutions belongs to the space H1(div, V) , i.e. the space of H1 functions whose divergence belongs to H1. The proposed mathematical frame is essential to conceptually base, on solid grounds, the numerical integration schemes required to investigate the properties of dilatational strain gradient elastic bodies. Their energy, as studied in the present paper, has manifold interests. Mathematically speaking, its singularity causes interesting mathematical difficulties whose overcoming leads to an increased understanding of the theory of second gradient continua. On the other hand, from the mechanical point of view, it gives an example of energy for a second gradient continuum which can sustain externally applied surface forces and double forces but cannot sustain externally applied surface couples. In this way, it is proven that couple stress continua, introduced by Toupin, represent only a particular case of the more general class of second gradient continua. Moreover, it is easily checked that for dilatational strain gradient continua, balance of force and balance of torques (or couples) are not enough to characterise equilibrium: to this aim, externally applied surface double forces must also be specified. As a consequence, the postulation scheme based on variational principles seems more suitable to study second gradient continua. It has to be remarked finally that dilatational strain gradient seems suitable to model the experimentally observed behaviour of some material used in 3D printing process
Time-fractional Cahn-Hilliard equation: Well-posedness, degeneracy, and numerical solutions
In this paper, we derive the time-fractional Cahn-Hilliard equation from
continuum mixture theory with a modification of Fick's law of diffusion. This
model describes the process of phase separation with nonlocal memory effects.
We analyze the existence, uniqueness, and regularity of weak solutions of the
time-fractional Cahn-Hilliard equation. In this regard, we consider
degenerating mobility functions and free energies of Landau, Flory--Huggins and
double-obstacle type. We apply the Faedo-Galerkin method to the system, derive
energy estimates, and use compactness theorems to pass to the limit in the
discrete form. In order to compensate for the missing chain rule of fractional
derivatives, we prove a fractional chain inequality for semiconvex functions.
The work concludes with numerical simulations and a sensitivity analysis
showing the influence of the fractional power. Here, we consider a convolution
quadrature scheme for the time-fractional component, and use a mixed finite
element method for the space discretization
Geometric partial differential equations: Surface and bulk processes
The workshop brought together experts representing a wide range of topics in geometric partial differential equations ranging from analyis over numerical simulation to real-life applications. The main themes of the conference were the analysis of curvature energies, new developments in pdes on surfaces and the treatment of coupled bulk/surface problems
On the well posedness of static boundary value problem within the linear dilatational strain gradient elasticity
AbstractIn this paper, it is proven an existence and uniqueness theorem for weak solutions of the equilibrium problem for linear isotropic dilatational strain gradient elasticity. Considered elastic bodies have as deformation energy the classical one due to Lamé but augmented with an additive term that depends on the norm of the gradient of dilatation: only one extra second gradient elastic coefficient is introduced. The studied class of solids is therefore related to Korteweg or Cahn–Hilliard fluids. The postulated energy naturally induces the space in which the aforementioned well-posedness result can be formulated. In this energy space, the introduced norm does involve the linear combination of some specific higher-order derivatives only: it is, in fact, a particular example of anisotropic Sobolev space. It is also proven that aforementioned weak solutions belongs to the space
H
1
(
d
i
v
,
V
)
, i.e. the space of
H
1
functions whose divergence belongs to
H
1
. The proposed mathematical frame is essential to conceptually base, on solid grounds, the numerical integration schemes required to investigate the properties of dilatational strain gradient elastic bodies. Their energy, as studied in the present paper, has manifold interests. Mathematically speaking, its singularity causes interesting mathematical difficulties whose overcoming leads to an increased understanding of the theory of second gradient continua. On the other hand, from the mechanical point of view, it gives an example of energy for a second gradient continuum which can sustain externally applied surface forces and double forces but cannot sustain externally applied surface couples. In this way, it is proven that couple stress continua, introduced by Toupin, represent only a particular case of the more general class of second gradient continua. Moreover, it is easily checked that for dilatational strain gradient continua, balance of force and balance of torques (or couples) are not enough to characterise equilibrium: to this aim, externally applied surface double forces must also be specified. As a consequence, the postulation scheme based on variational principles seems more suitable to study second gradient continua. It has to be remarked finally that dilatational strain gradient seems suitable to model the experimentally observed behaviour of some material used in 3D printing process
Second-order flows for approaching stationary points of a class of non-convex energies via convex-splitting schemes
The use of accelerated gradient flows is an emerging field in optimization, scientific computing and beyond. This paper contributes to the theoretical underpinnings of a recently-introduced computational paradigm known as second-order flows, which demonstrate significant performance particularly for the minimization of non-convex energy functionals defined on Sobolev spaces, and are characterized by novel dissipative hyperbolic partial differential equations. Our approach hinges upon convex-splitting schemes, a tool which is not only pivotal for clarifying the well-posedness of second-order flows, but also yields a versatile array of robust numerical schemes through temporal and spatial discretization. We prove the convergence to stationary points of such schemes in the semi-discrete setting. Further, we establish their convergence to time-continuous solutions as the time-step tends to zero, and perform a comprehensive error analysis in the fully discrete case. Finally, these algorithms undergo thorough testing and validation in approaching stationary points of non-convex variational models in applied sciences, such as the Ginzburg-Landau energy in phase-field modeling and a specific case of the Landau-de Gennes energy of the Q-tensor model for liquid crystals
Second-order flows for approaching stationary points of a class of non-convex energies via convex-splitting schemes
The use of accelerated gradient flows is an emerging field in optimization,
scientific computing and beyond. This paper contributes to the theoretical
underpinnings of a recently-introduced computational paradigm known as
second-order flows, which demonstrate significant performance particularly for
the minimization of non-convex energy functionals defined on Sobolev spaces,
and are characterized by novel dissipative hyperbolic partial differential
equations. Our approach hinges upon convex-splitting schemes, a tool which is
not only pivotal for clarifying the well-posedness of second-order flows, but
also yields a versatile array of robust numerical schemes through temporal and
spatial discretization. We prove the convergence to stationary points of such
schemes in the semi-discrete setting. Further, we establish their convergence
to time-continuous solutions as the time-step tends to zero, and perform a
comprehensive error analysis in the fully discrete case. Finally, these
algorithms undergo thorough testing and validation in approaching stationary
points of non-convex variational models in applied sciences, such as the
Ginzburg-Landau energy in phase-field modeling and a specific case of the
Landau-de Gennes energy of the Q-tensor model for liquid crystals.Comment: 37 pages, 4 figure
On a Class of Sixth-order Cahn-Hilliard Type Equations with Logarithmic Potential
We consider a class of six-order Cahn-Hilliard equations with logarithmic
type potential. This system is closely connected with some important
phase-field models relevant in different applications, for instance, the
functionalized Cahn-Hilliard equation that describes phase separation in
mixtures of amphiphilic molecules in solvent, and the Willmore regularization
of Cahn-Hilliard equation for anisotropic crystal and epitaxial growth. The
singularity of the configuration potential guarantees that the solution always
stays in the physical relevant domain [-1,1]. Meanwhile, the resulting system
is characterized by some highly singular diffusion terms that make the
mathematical analysis more involved. We prove existence and uniqueness of
global weak solutions and show their parabolic regularization property for any
positive time. Besides, we investigate long-time behavior of the system,
proving existence of the global attractor for the associated dynamical process
in a suitable complete metric space.Comment: 35 page
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Emerging Developments in Interfaces and Free Boundaries
The field of the mathematical and numerical analysis of systems of nonlinear partial differential equations involving interfaces and free boundaries is a well established and flourishing area of research. This workshop focused on recent developments and emerging new themes. By bringing together experts in these fields we achieved progress in open questions and developed novel research directions in mathematics related to interfaces and free boundaries. This interdisciplinary workshop brought together researchers from distinct mathematical fields such as analysis, computation, optimisation and modelling to discuss emerging challenges
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