In this work we propose and analyze an abstract parameter dependent model written as a mixed variational formulation based on Volterra integrals of second kind. For the analysis, we consider a suitable adaptation to the classic mixed theory in the Volterra equations setting, and prove the well posedness of the resulting mixed viscoelastic formulation. Error estimates are derived, using the available results for Volterra equations, where all the estimates are independent of the perturbation parameter. We consider an application of the developed theory in a viscoelastic Timoshenko beam, and report numerical experiments in order to assess the independence of the perturbation parameter.
We introduce and analyze a new mixed finite element method with reduced symmetry for the standard linear model in viscoelasticity. Following a previous approach employed for linear elastodynamics, the present problem is formulated as a second-order hyperbolic partial differential equation in which, after using the motion equation to eliminate the displacement unknown, the stress tensor remains as the main variable to be found. The resulting variational formulation is shown to be well-posed, and a class of $\text{H}(\text{div})$-conforming semi-discrete schemes is proved to be convergent. Then,…
A virtual element method for Kelvin--Voigt viscoelasticity
Monash University（莫纳什大学）；Universidad Adventista de Chile（智利基督复临安息日会大学）；Indian Institute of Space Science and Technology（印度空间科学技术研究所）；Birla Institute of Technology and Science, Pilani（比拉理工与科学学院（皮拉尼校区））
This study provides an abstract framework to analyze mixed formulations in viscoelasticity, in the classic saddle point form. Standard hypothesis for mixed methods are adapted to the Volterra type equations in order to obtain stability of the proposed problem. Error estimates are derived for suitable finite element spaces. We apply the developed theory to a bending moment formulation for a linear viscoelastic Timoshenko beam and for the Laplace operator with memory terms. For both problems we report numerical results to asses the performance of the methods.
We introduce and analyze a stress-based formulation for Zener's model in linear viscoelasticity. The method is aimed to tackle efficiently heterogeneous materials that admit purely elastic and viscoelastic parts in their composition. We write the mixed variational formulation of the problem in terms of a class of tensorial wave equation and obtain an energy estimate that guaranties the well-posedness of the problem through a standard Galerkin procedure. We propose and analyze mixed continuous and discontinuous Galerkin space discretizations of the problem and derive optimal error bounds for ea…
National Center for Supercomputing Applications（国家超级计算应用中心）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Advanced Digital & Additive Manufacturing Center（先进数字与增材制造中心）；Khalifa University of Science and Technology（哈利法科学技术大学）
The potential energy formulation and deep learning are merged to solve partial differential equations governing the deformation in hyperelastic and viscoelastic materials. The presented deep energy method (DEM) is self-contained and meshfree. It can accurately capture the three-dimensional (3D) mechanical response without requiring any time-consuming training data generation by classical numerical methods such as the finite element method. Once the model is appropriately trained, the response can be attained almost instantly at any point in the physical domain, given its spatial coordinates. T…
We study a non standard mixed formulation of the Poisson problem, sometimes known as dual mixed formulation. For reasons related to the equilibration of the flux, we use finite elements that are conforming in H(div) for the approximation of the gradients, even if the formulation would allow for discontinuous finite elements. The scheme is not uniformly inf-sup stable, but we can show existence and uniqueness of the solution, as well as optimal error estimates for the gradient variable when suitable regularity assumptions are made. Several additional remarks complete the paper, shedding some li…
We develop a mixed finite element method for the coupled problem arising in the interaction between a free fluid governed by the Stokes equations and flow in deformable porous medium modeled by the Biot system of poroelasticity. Mass conservation, balance of stress, and the Beavers--Joseph--Saffman condition are imposed on the interface. We consider a fully mixed Biot formulation based on a weakly symmetric stress-displacement-rotation elasticity system and Darcy velocity-pressure flow formulation. A velocity-pressure formulation is used for the Stokes equations. The interface conditions are i…
We analyze Galerkin discretizations of a new well-posed mixed space-time variational formulation of parabolic PDEs. For suitable pairs of finite element trial spaces, the resulting Galerkin operators are shown to be uniformly stable. The method is compared to two related space-time discretization methods introduced in [IMA J. Numer. Anal., 33(1) (2013), pp. 242-260] by R. Andreev and in [Comput. Methods Appl. Math., 15(4) (2015), pp. 551-566] by O. Steinbach.
In the present paper we describe the computational implementation of some integral terms that arise from mixed virtual element methods (mixed-VEM) in two-dimensional pseudostress-velocity formulations. The implementation presented here consider any polynomial degree $k \geq 0$ in a natural way by building several local matrices of small size through the matrix multiplication and the Kronecker product. In particular, we apply the foregoing mentioned matrices to the Navier-Stokes equations with Dirichlet boundary conditions, whose mixed-VEM formulation was originally proposed and analyzed in a r…
We propose and analyze a mixed formulation for the Brinkman-Forchheimer equations for unsteady flows. Our approach is based on the introduction of a pseudostress tensor related to the velocity gradient, leading to a mixed formulation where the pseudostress tensor and the velocity are the main unknowns of the system. We establish existence and uniqueness of a solution to the weak formulation in a Banach space setting, employing classical results on nonlinear monotone operators and a regularization technique. We then present well-posedness and error analysis for semidiscrete continuous-in-time a…
In this paper, we consider an acoustic wave transmission problem with mixed boundary conditions of Dirichlet, Neumann, and impedance type. The transmission interfaces may join the domain boundary in a general way independent of the location of the boundary conditions. We will derive a formulation as a \textit{direct}, \textit{space-time retarded boundary integral equation}, where both Cauchy data are kept as unknowns on the impedance part of the boundary. This requires the definition of single-trace spaces which incorporate homogeneous Dirichlet and Neumann conditions on the corresponding part…
Optimal block preconditioners for a mass-conserving mixed stress formulation of Stokes flow
Mathematical Institute, University of Oxford（牛津大学数学研究所）；School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）理工学院）
In two and three dimensional domains, we analyze mixed finite element methods for a velocity-pressure-pseudostress formulation of the Stokes eigenvalue problem. The methods consist in two schemes: the velocity and pressure are approximated with piecewise polynomial and for the pseudostress we consider two classic families of finite elements for $\mathrm{H}(\mathop{\mathrm{div}})$ spaces: the Raviart-Thomas and the Brezzi-Douglas Marini elements. With the aid of the classic spectral theory for compact operators, we prove that our method does not introduce spurious modes. Also, we obtain converg…
Unfitted finite element methods have emerged as a popular alternative to classical finite element methods for the solution of partial differential equations and allow modeling arbitrary geometries without the need for a boundary-conforming mesh. On the other hand, the efficient solution of the resultant system is a challenging task because of the numerical ill-conditioning that typically entails from the formulation of such methods. We use an adaptive geometric multigrid solver for the solution of the mixed finite cell formulation of saddle-point problems and investigate its convergence in the…
Discrete Fracture and Matrix (DFM) models describe fractured porous media as complex sets of 2D planar polygons embedded in a 3D matrix representing the surrounding porous medium. The numerical simulation of the flow in a DFM requires the discretization of partial differential equations on the three dimensional matrix, the planar fractures and the one dimensional fracture intersections, and suitable coupling conditions between entities of different dimensionality need to be added at the various interfaces to close the problem. The present work proposes an arbitrary order implementation of the …
In this paper, we develop the constraint energy minimization generalized multiscale finite element method (CEM-GMsFEM) in mixed formulation applied to parabolic equations with heterogeneous diffusion coefficients. The construction of the method is based on two multiscale spaces: pressure multiscale space and velocity multiscale space. The pressure space is constructed via a set of well-designed local spectral problems, which can be solved independently. Based on the computed pressure multiscale space, we will construct the velocity multiscale space by applying constrained energy minimization. …
We present a mixed finite element method for a five-field formulation of the Biot system of poroelasticity that reduces to a cell-centered pressure-displacement system on simplicial and quadrilateral grids. A mixed stress-displacement-rotation formulation for elasticity with weak stress symmetry is coupled with a mixed velocity-pressure Darcy formulation. The spatial discretization is based on combining the multipoint stress mixed finite element (MSMFE) method for elasticity and the multipoint flux mixed finite element (MFMFE) method for Darcy flow. It uses the lowest order Brezzi-Douglas-Mari…
We propose two different discrete formulations for the weak imposition of the Neumann boundary conditions of the Darcy flow. The Raviart-Thomas mixed finite element on both triangular and quadrilateral meshes is considered for both methods. One is a consistent discretization depending on a weighting parameter scaling as $\mathcal O(h^{-1})$, while the other is a penalty-type formulation obtained as the discretization of a perturbation of the original problem and relies on a parameter scaling as $\mathcal O(h^{-k-1})$, $k$ being the order of the Raviart-Thomas space. We rigorously prove that bo…
Conforming/Non-Conforming Mixed Finite Element Methods for Optimal Control of Velocity-Vorticity-Pressure Formulation for the Oseen Problem with Variable Viscosity
Space-time Domain Decomposition and Mixed Formulation for solving reduced fracture models
INRIA Paris-Rocquencourt（法国国家信息与自动化研究所巴黎-罗盖特分校）；Ho Chi Minh City University of Pedagogy（胡志明市师范大学）；Université Paris 13, UMR 7539, LAGA（巴黎第十三大学，UMR 7539，LAGA实验室）
Decoupling of Mixed Methods Based on Generalized Helmholtz Decompositions
Department of Mathematics, University of California at Irvine（加州大学尔湾分校数学系）；School of Mathematics, Shanghai University of Finance and Economics（上海财经大学数学学院）
In this work, we consider fracture propagation in nearly incompressible and (fully) incompressible materials using a phase-field formulation. We use a mixed form of the elasticity equation to overcome volume locking effects and develop a robust, nonlinear and linear solver scheme and preconditioner for the resulting system. The coupled variational inequality system, which is solved monolithically, consists of three unknowns: displacements, pressure, and phase-field. Nonlinearities due to coupling, constitutive laws, and crack irreversibility are solved using a combined Newton algorithm for the…
