اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدAnalysis of Fully Discrete Finite Element Methods for a System of Differential Equations Modeling Swelling Dynamics of Polymer Gels
354
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
The goal of this paper is to develop and analyze some fully discrete finite element methods for a displacement-pressure model modeling swelling dynamics of polymer gels under mechanical constraints. In the model, the swelling dynamics is governed by the solvent permeation and the elastic interaction; the permeation is described by a pressure equation for the solvent, and the elastic interaction is described by displacement equations for the solid network of the gel. By introducing an elastic pressure we first present a reformulation of the original model, and then propose a time-stepping scheme which decouples the PDE system at each time step into two sub-problems, one of which is a generalized Stokes problem for the displacement vector field and another is a diffusion problem for a pseudo-pressure field. To make such a multiphysical approach feasible, it is vital to discover admissible constraints to resolve the uniqueness issue for both sub-problems. The main advantage of the proposed approach is that it allows one to utilize any convergent Stokes solver together with any convergent diffusion equation solver to solve the polymer gel model. In the paper, the Taylor-Hood mixed finite element method combined with the continuous linear finite element method are used as an example to present the ideas and to demonstrate the viability of the proposed multiphysical approach. It is proved that, under a mesh constraint, both the proposed semi-discrete (in space) and fully discrete methods enjoy some discrete energy laws which mimic the differential energy law satisfied by the PDE solution. Optimal order error estimates in various norms are established for the numerical solutions of both the semi-discrete and fully discrete methods. Numerical experiments are also presented to show the efficiency of the proposed approach and methods.
قيم البحث
اقرأ أيضاً
Polyelectrolyte gels are a very attractive class of actuation materials with remarkable electronic and mechanical properties with a great similarity to biological contractile tissues. They consist of a polymer network with ionizable groups and a liqu
id phase with mobile ions. Absorption and delivery of solvent lead to a large change of volume. This mechanism can be triggered by chemical (change of salt concentration or pH of solution surrounding the gel), electrical, thermal or optical stimuli. Due to this capability, these gels can be used as actuators for technical applications, where large swelling and shrinkage is desired. In the present work chemically stimulated polymer gels in a solution bath are investigated. To adequately describe the different complicated phenomena occurring in these gels, they can be modeled on different scales. Therefore, models based on the statistical theory and porous media theory, as well as a coupled multi-field model and a discrete element formulation are derived and employed. In this paper, the coupled multi-field model and the discrete element model for chemical stimulation of a polymer gel film with and without domain deformation are employed. Based on these results, the presented formulations are compared and conclusions on their applicability in engineering practice are finally drawn.
In this paper, we define new unfitted finite element methods for numerically approximating the solution of surface partial differential equations using bulk finite elements. The key idea is that the $n$-dimensional hypersurface, $Gamma subset mathbb{
R}^{n+1}$, is embedded in a polyhedral domain in $mathbb R^{n+1}$ consisting of a union, $mathcal{T}_h$, of $(n+1)$-simplices. The finite element approximating space is based on continuous piece-wise linear finite element functions on $mathcal{T}_h$. Our first method is a sharp interface method, emph{SIF}, which uses the bulk finite element space in an approximating weak formulation obtained from integration on a polygonal approximation, $Gamma_{h}$, of $Gamma$. The full gradient is used rather than the projected tangential gradient and it is this which distinguishes emph{SIF} from the method of [42]. The second method, emph{NBM}, is a narrow band method in which the region of integration is a narrow band of width $O(h)$. emph{NBM} is similar to the method of [13]. but again the full gradient is used in the discrete weak formulation. The a priori error analysis in this paper shows that the methods are of optimal order in the surface $L^{2}$ and $H^{1}$ norms and have the advantage that the normal derivative of the discrete solution is small and converges to zero. Our third method combines bulk finite elements, discrete sharp interfaces and narrow bands in order to give an unfitted finite element method for parabolic equations on evolving surfaces. We show that our method is conservative so that it preserves mass in the case of an advection diffusion conservation law. Numerical results are given which illustrate the rates of convergence.
The paper is concerned with the adaptive finite element solution of linear elliptic differential equations using equidistributing meshes. A strategy is developed for defining this type of mesh based on residual-based a posteriori error estimates and
rigorously analyzing the convergence of a linear finite element approximation using them. The existence and computation of equidistributing meshes and the continuous dependence of the finite element approximation on mesh are also studied. Numerical results are given to verify the theoretical findings.
Sparse spectral methods for solving partial differential equations have been derived in recent years using hierarchies of classical orthogonal polynomials on intervals, disks, and triangles. In this work we extend this methodology to a hierarchy of n
on-classical orthogonal polynomials on disk slices (e.g. a half-disk) and trapeziums. This builds on the observation that sparsity is guaranteed due to the boundary being defined by an algebraic curve, and that the entries of partial differential operators can be determined using formulae in terms of (non-classical) univariate orthogonal polynomials. We apply the framework to solving the Poisson, variable coefficient Helmholtz, and Biharmonic equations.
In this paper, we introduce and analyse a surface finite element discretization of advection-diffusion equations with uncertain coefficients on evolving hypersurfaces. After stating unique solvability of the resulting semi-discrete problem, we prove
optimal error bounds for the semi-discrete solution and Monte Carlo samplings of its expectation in appropriate Bochner spaces. Our theoretical findings are illustrated by numerical experiments in two and three space dimensions.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
