1,721,076 research outputs found
Error analysis of a finite element approximation of a degenerate Cahn-Hilliard equation
Full text linkThis work considers a Cahn-Hilliard type equation with degenerate mobility and single-well potential of Lennard-Jones type, motivated by increasing interest in diffuse interface modelling of solid tumors. The degeneracy set of the mobility and the singularity set of the potential do not coincide, and the zero of the potential is an unstable equilibrium configuration. This feature introduces a nontrivial difference with respect to the Cahn-Hilliard equation analyzed in the literature. In particular, the singularities of the potential do not compensate the degeneracy of the mobility by constraining the solution to be strictly separated from the degeneracy values. The error analysis of a well posed continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality, is developed. Whilst in previous works the error analysis of suitable finite element approximations has been studied for second order degenerate and fourth order non degenerate parabolic equations, in this work the a priori estimates of the error between the discrete solution and the weak solution to which it converges are obtained for a degenerate fourth order parabolic equation. The theoretical error estimates obtained in the present case state that the norms of the approximation errors, calculated on the support of the solution in the proper functional spaces, are bounded by power laws of the discretization parameters with exponent 1/2, while in the case of the classical Cahn-Hilliard equation with constant mobility the exponent is 1. The estimates are finally succesfully validated by simulation results in one and two space dimensions
Discontinuous Galerkin finite element discretization of a degenerate Cahn–Hilliard equation with a single-well potential
No full textThis work concerns the analysis of a discontinuous Galerkin finite element approximation of a degenerate Cahn–Hilliard equation with single-well potential of the Lennard-Jones type. This equation is widely used for the diffuse interface modeling of solid tumors. A finite element discretization with discontinuous elements of the problem is developed, where the positivity of the solution, which is not straightforwardly guaranteed at the discrete level, is enforced through a variational inequality. The well posedness of the formulation is shown, together with the convergence to the weak solution. This discretization properly selects the solutions with a moving support with finite velocity, discarding the unphysical solutions with fixed support. The simulation results in two space dimensions are reported to test the validity of the proposed algorithm. Similar results as the ones obtained in standard phase ordering dynamics are found, highlighting the evolution of single domains to steady state with constant curvature. Imposing known solutions, good convergence properties of the discrete solution to the continuous one are observed using a proper norm calculated on its support. Contrarily to the discretizations with continuous elements, the use of discontinuous elements aims at recovering the optimal convergence rate found in literature for the finite element approximations of the Cahn–Hilliard equation with constant mobility. It is also useful when dealing with the dynamics of domains with corners, with highly heterogeneous materials and in presence of advective terms
Il Corso di laurea in Scienze dell’educazione. Una ricerca fra gli iscritti di Verona
No full textUna ricerca condotta fra gli iscritti del corso di laurea in Scienze dell'Educazione di Verona per conoscere la loro formazione precedente, l'impatto con la realtà universitaria ed i loro orientamenti futuri
STRICT SEPARATION AND NUMERICAL APPROXIMATION FOR A NON-LOCAL CAHN-HILLIARD EQUATION WITH SINGLE-WELL POTENTIAL
No full textIn this paper we study a non-local Cahn-Hilliard equation with singular single-well potential and degenerate mobility. This results as a particular case of a more general model derived for a binary, saturated, closed and incompressible mixture, composed by a tumor phase and a healthy phase, evolving in a bounded domain. The general system couples a Darcy-type evolution for the average velocity field with a convective reaction-diffusion type evolution for the nutrient concentration and a non-local convective Cahn-Hilliard equation for the tumor phase. The main mathematical difficulties are related to the proof of the separation property for the tumor phase in the Cahn-Hilliard equation: up to our knowledge, such problem is indeed open in the literature. For this reason, in the present contribution we restrict the analytical study to the Cahn-Hilliard equation only. For the non-local Cahn-Hilliard equation with singular single-well potential and degenerate mobility, we study the existence and uniqueness of weak solutions for spatial dimensions d <= 3. After showing existence, we prove the strict separation property in three spatial dimensions, implying the same property also for lower spatial dimensions, which opens the way to the proof of uniqueness of solutions. Finally, we propose a well posed and gradient stable continuous finite element approximation of the model for d <= 3, which preserves the physical properties of the continuous solution and which is computationally efficient, and we show simulation results in two spatial dimensions which prove the consistency of the proposed scheme and which describe the phase ordering dynamics associated to the system
The constitutive relations of initially stressed incompressible Mooney-Rivlin materials
No full textInitial stresses originate in soft materials by the occurrence of misfits in the undeformed microstructure. Since the reference configuration is not stress-free, the effects of initial stresses on the hyperelastic behavior must be constitutively addressed. Notably, the free energy of an initially stressed material may not possess the same symmetry group as the one of the same material deforming from a naturally unstressed configuration. This work assumes that the hyperelastic strain energy density is characterized only by the deformation gradient and the initial stress tensor, using an explicit functional dependence on their independent invariants. In particular, we consider a subclass of constitutive behaviors in which the material constants do not depend on the choice of the reference configuration. Within this theoretical framework, a constitutive equation is derived for an initially stressed body that naturally behaves as an incompressible Mooney-Rivlin material. The strain energy densities for initially stressed neo-Hookean and Mooney materials are derived as special sub-cases. By assuming the existence of a virtual state that is naturally stress-free, the resulting strain energy functions are proved to fulfill the required frame-independence constraints for this special class of constitutive models. In the case of plane strain, great simplifications arise in the expression of the constitutive relations. Finally, the resulting constitutive relations prove useful guidelines for designing non-destructive methods for the quantification of the underlying initial stresses in naturally isotropic materials
Analysis of a model for precipitation and dissolution coupled with a Darcy flux
No full textIn this paper we deal with the numerical analysis of an upscaled model of a reactive flow in a porous medium, which describes the transport of solutes undergoing precipitation and dissolution, leading to the formation/degradation of crystals inside the porous matrix. The model is defined at the Darcy scale, and it is coupled to a Darcy flow characterized by a permeability field that changes in space and time according to the precipitated crystal concentration. The model involves a non-linear multi-valued reaction term, which is treated exactly by solving an inclusion problem for the solutes and the crystals dynamics. We consider a weak formulation for the coupled system of equations expressed in a dual mixed form for the Darcy field and in a primal form for the solutes and the precipitate, and show its well posedness without resorting to regularization of the reaction term. Convergence to the weak solution is proved for its finite element approximation. We perform numerical experiments to study the behavior of the system and to assess the effectiveness of the proposed discretization strategy. In particular we show that a method that captures the discontinuity yields sharper dissolution fronts with respect to methods that regularize the discontinuous term
On the existence of elastic minimizers for initially stressed materials
No full textA soft solid is said to be initially stressed if it is subjected to a state of internal stress in its unloaded reference configuration. In physical terms, its stored elastic energy may not vanish in the absence of an elastic deformation, being also dependent on the spatial distribution of the underlying material inhomogeneities. Developing a sound mathematical framework to model initially stressed solids in nonlinear elasticity is key for many applications in engineering and biology. This work investigates the links between the existence of elastic minimizers and the constitutive restrictions for initially stressed materials subjected to finite deformations. In particular, we consider a subclass of constitutive responses in which the strain energy density is taken as a scalar-valued function of both the deformation gradient and the initial stress tensor. The main advantage of this approach is that the initial stress tensor belongs to the group of divergence-free symmetric tensors satisfying the boundary conditions in any given reference configuration. However, it is still unclear which physical restrictions must be imposed for the well-posedness of this elastic problem. Assuming that the constitutive response depends on the choice of the reference configuration only through the initial stress tensor, under given conditions we prove the local existence of a relaxed state given by an implicit tensor function of the initial stress distribution. This tensor function is generally not unique, and can be transformed according to the symmetry group of the material at fixed initial stresses. These results allow one to extend Ball's existence theorem of elastic minimizers for the proposed constitutive choice of initially stressed materials
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