On the convergence of a regularization scheme for approximating cavitation solutions with prescribed cavity volume size

Let $\Omega\in\mathbb{R}^n$, $n=2,3$, be the region occupied by a hyperelastic body in its reference configuration. Let $E(\cdot)$ be the stored energy functional, and let $x_0$ be a flaw point in $\Omega$ (i.e., a point of possible discontinuity for admissible deformations of the body). For V>0 fixed, let $u_V$ be a minimizer of $E(\cdot)$ among the set of discontinuous deformations $u$ constrained to form a hole of prescribed volume $V$ at $x_0$ and satisfying the homogeneous boundary data $u(x)=Ax$ for $x\in\partial \Omega$. In this paper we describe a regularization scheme for the computation of both $u_V$ and $E(u_V)$ and study its convergence properties. In particular, we show that as the regularization parameter goes to zero, (a subsequence) of the regularized constrained minimizers converge weakly in $W^{1,p}(\Omega\setminus{{\mathcal{B}}_{\delta}(x_0)})$ to a minimizer $u_{V}$ for any \delta>0. We obtain various sensitivity results for the dependence of the energies and Lagrange multipliers of the regularized constrained minimizers on the boundary data $A$ and on the volume parameter $V$. We show that both the regularized constrained minimizers and $u_V$ satisfy suitable weak versions of the corresponding Euler--Lagrange equations. In addition we describe the main features of a numerical scheme for approximating $u_V$ and $E(u_V)$ and give numerical examples for the case of a stored energy function of an elastic fluid and in the case of the incompressible limit

On the convergence of a regularization scheme for approximating cavitation solutions with prescribed cavity volume size

Let $\Omega\in\mathbb{R}^n$, $n=2,3$, be the region occupied by a hyperelastic body in its reference configuration. Let $E(\cdot)$ be the stored energy functional, and let $x_0$ be a flaw point in $\Omega$ (i.e., a point of possible discontinuity for admissible deformations of the body). For V>0 fixed, let $u_V$ be a minimizer of $E(\cdot)$ among the set of discontinuous deformations $u$ constrained to form a hole of prescribed volume $V$ at $x_0$ and satisfying the homogeneous boundary data $u(x)=Ax$ for $x\in\partial \Omega$. In this paper we describe a regularization scheme for the computation of both $u_V$ and $E(u_V)$ and study its convergence properties. In particular, we show that as the regularization parameter goes to zero, (a subsequence) of the regularized constrained minimizers converge weakly in $W^{1,p}(\Omega\setminus{{\mathcal{B}}_{\delta}(x_0)})$ to a minimizer $u_{V}$ for any \delta>0. We obtain various sensitivity results for the dependence of the energies and Lagrange multipliers of the regularized constrained minimizers on the boundary data $A$ and on the volume parameter $V$. We show that both the regularized constrained minimizers and $u_V$ satisfy suitable weak versions of the corresponding Euler--Lagrange equations. In addition we describe the main features of a numerical scheme for approximating $u_V$ and $E(u_V)$ and give numerical examples for the case of a stored energy function of an elastic fluid and in the case of the incompressible limit

Cavitation of a spherical body under mechanical and self-gravitational forces

In this paper, we look for minimizers of the energy functional for isotropic compressible elasticity taking into consideration the effect of a gravitational field induced by the body itself. We consider two types of problems: the displacement problem in which the outer boundary of the body is subjected to a Dirichlet-type boundary condition, and the one with zero traction on the boundary but with an internal pressure function. For a spherically symmetric body occupying the unit ball, the minimization is done within the class of radially symmetric deformations. We give conditions for the existence of such minimizers, for satisfaction of the Euler-Lagrange equations, and show that for large displacements or large internal pressures, the minimizer must develop a cavity at the centre. We discuss a numerical scheme for approximating the minimizers for the displacement problem, together with some simulations that show the dependence of the cavity radius and minimum energy on the displacement and mass density of the body.</p

Cavitation of a spherical body under mechanical and self-gravitational forces

In this paper, we look for minimizers of the energy functional for isotropic compressible elasticity taking into consideration the effect of a gravitational field induced by the body itself. We consider two types of problems: the displacement problem in which the outer boundary of the body is subjected to a Dirichlet-type boundary condition, and the one with zero traction on the boundary but with an internal pressure function. For a spherically symmetric body occupying the unit ball, the minimization is done within the class of radially symmetric deformations. We give conditions for the existence of such minimizers, for satisfaction of the Euler-Lagrange equations, and show that for large displacements or large internal pressures, the minimizer must develop a cavity at the centre. We discuss a numerical scheme for approximating the minimizers for the displacement problem, together with some simulations that show the dependence of the cavity radius and minimum energy on the displacement and mass density of the body.</p

Infinite energy cavitating solutions: a variational approach

We study the phenomenon of cavitation for the displacement boundary value problem of radial, isotropic compressible elasticity for a class of stored energy functions of the form $W(F) + h(\det F)$, where $W$ grows like $||F||^n$, and $n$ is the space dimension. In this case it follows (from a result of Vodopyanov, Goldshtein and Reshetnyak) that discontinuous deformations must have infinite energy. After characterizing the rate at which this energy blows up, we introduce a modified energy functional which differs from the original by a null lagrangian, and for which cavitating energy minimizers with finite energy exist. In particular, the Euler--Lagrange equations for the modified energy functional are identical to those for the original problem except for the boundary condition at the inner cavity. This new boundary condition states that a certain modified radial Cauchy stress function has to vanish at the inner cavity. This condition corresponds to the radial Cauchy stress for the original functional diverging to $-\infty$ at the cavity surface. Many previously known variational results for finite energy cavitating solutions now follow for the modified functional, such as the existence of radial energy minimizers, satisfaction of the Euler-Lagrange equations for such minimizers, and the existence of a critical boundary displacement for cavitation. We also discuss a numerical scheme for computing these singular cavitating solutions using regular solutions for punctured balls. We show the convergence of this numerical scheme and give some numerical examples including one for the incompressible limit case. Our approach is motivated in part by the use of the renormalized energy for Ginzberg-Landau vortices.Comment: 23 pages, 4 figure

Variation of the Chlorophyll a Related to Sea Surface Temperature, Wind and Geostrophic Currents in the Cape Verde Region Using Satellite Data

We present a comparative analysis of satellite derived climatologies in the Cape Verde region (CV). In order to establish chlorophyll a variability, in relation to other oceanographic phenomena, a set of, relatively long (from five to eight years), time series of chlorophyll a, sea surface temperature, wind and geostrophic currents, were ensembled for the Eastern Central Atlantic (ECA). We studied seasonal and inter-annual variability of phytoplankton concentration, in relation to the rest of the variables, with a special focus in CV. We compared the situation within the archipelago with those of the surrounding marine environments, such as the North West African Upwelling (NWAU), North Atlantic Subtropical Gyre (NASTG), North Equatorial Counter Current (NECC) and Guinea Dome (GD). At the seasonal scale, CV region behaves partly as the surrounding areas, nevertheless, some autochthonous features were also found. The maximum peak of the pigment having a positive correlation with temperature is found at the end of the year for all the points in the archipelago; a less remarkable rise with negative correlation is also detected in February for points CV2 and CV4. This is behavior that none of the surrounding environments have shown. This enrichment was found to be preceded by a drastic drop in wind intensity (SW Monsoon) during summer months. The inter-annual analysis shows a tendency for decreasing of the chlorophyll a concentration.Utilizando séries temporais (entre cinco e oito anos) de dados de satélite a grande escala para a zona de Cabo Verde (CV), faz-se uma análise da variabilidade da clorofila a relacionando-a com outros parâmetros oceanográficos como a temperatura superficial do mar, o vento e as correntes geostróficas. Estuda-se a variabilidade estacional e interanual da concentração do fitoplancton em relação ao resto das variáveis comparando a situação nas águas de Cabo Verde com o ambiente marinho à volta do arquipélago como o Upwelling Nordeste Africano (NWAU), o Giro Subtropical Norte-Atlântico (NASTG), a Contra Corrente Norte-Equatorial (NECC) e o Domo da Guiné (GD). À escala estacional, a zona de Cabo Verde comporta-se como parte das regiões envolventes, no entanto, algumas características autóctonas foram também encontradas. O pico máximo do pigmento mostrando uma correlação positiva com a temperatura foi encontrado no final do ano em todos os pontos eulerianos definidos para o arquipélago; um incremento menos notável, e com uma correlação negativa, também é detectada nos pontos CV2 e CV4. Este comportamento não foi visto em nenhum dos pontos do ambiente circundante ao arquipélago. O enriquecimento no final do ano foi precedido por um drástico decréscimo na intensidade do vento (Monsão do SW) durante os meses de verão. A análise interanual mostra uma tendência para o decréscimo da concentração da clorofila a