Search

We introduce an energy stable, high-order-accurate finite difference approximation of the dynamic, pure bending Kirchhoff plate equations for complex geometries and spatially variable properties. We utilize the summation-by-parts (SBP) framework to discretize the biharmonic operator with variable coefficients, with attention given to free and clamped boundary conditions and corner conditions. Energy conservation is established by combining SBP boundary closures with weak enforcement of the boundary and interface conditions using a penalty (simultaneous approximation term, SAT) technique. Then we couple the plate equations to the shallow water equations to study flexural-gravity wave propagation, and prove that the semi-discrete system of equations is self-adjoint. We demonstrate the stability and accuracy properties of the method on curvilinear multiblock grids using the method of manufactured solutions. The method, which we provide in an open-source code, is then used to model ocean wave interactions with the Thwaites Glacier and Pine Island Ice Shelf in the Amundsen Sea off the coast of West Antarctica., eSSENCE - An eScience Collaboration

In this paper, the blow-up of solutions for a Dirichlet-Neumann problem to initial nonlinear viscoelastic plate equation with a lower order perturbation of p(x,t)-Laplacian operator in the presence of time delay is obtained. Under suitable conditions on g and the variable exponent of the p(x,t)-Laplacian operator, we prove that any weak solution with nonpositive initial energy as well as positive initial energy blows up in a finite time.

In this paper, the blow-up of solutions for a Dirichlet-Neumann problem to initial nonlinear viscoelastic plate equation with a lower order perturbation of p(x,t)-Laplacian operator in the presence of time delay is obtained. Under suitable conditions on g and the variable exponent of the p(x,t)-Laplacian operator, we prove that any weak solution with nonpositive initial energy as well as positive initial energy blows up in a finite time.

In this paper, the blow-up of solutions for a Dirichlet-Neumann problem to initial nonlinear viscoelastic plate equation with a lower order perturbation of p(x,t)-Laplacian operator in the presence of time delay is obtained. Under suitable conditions on g and the variable exponent of the p(x,t)-Laplacian operator, we prove that any weak solution with nonpositive initial energy as well as positive initial energy blows up in a finite time.

We study the high-contrast biharmonic plate equation with Hsieh-Clough-Tocher discretization. We construct a preconditioner that is robust with respect to contrast size and mesh size simultaneously based on the preconditioner proposed by Aksoylu et al. (Comput. Vis. Sci. 2008; 11: 319-331). By extending the devised singular perturbation analysis from linear finite element discretization to the above discretization, we prove and numerically demonstrate the robustness of the preconditioner. Therefore, we accomplish a desirable preconditioning design goal by using the same family of preconditioners to solve the elliptic family of PDEs with varying discretizations. We also present a strategy on how to generalize the proposed preconditioner to cover high-contrast elliptic PDEs of order 2k, k>2. Moreover, we prove a fundamental qualitative property of the solution to the high-contrast biharmonic plate equation. Namely, the solution over the highly bending island becomes a linear polynomial asymptotically. The effectiveness of our preconditioner is largely due to the integration of this qualitative understanding of the underlying PDE into its construction. Copyright (C) 2010 John Wiley ; Sons, Ltd., NSFNational Science Foundation (NSF) [DMS-1016190]; Direct For Mathematical ; Physical ScienNational Science Foundation (NSF)NSF - Directorate for Mathematical ; Physical Sciences (MPS) [1016190] Funding Source: National Science Foundation, This work is supported by NSF under grant number DMS-1016190.

We study the high-contrast biharmonic plate equation with Hsieh-Clough-Tocher discretization. We construct a preconditioner that is robust with respect to contrast size and mesh size simultaneously based on the preconditioner proposed by Aksoylu et al. (Comput. Vis. Sci. 2008; 11: 319-331). By extending the devised singular perturbation analysis from linear finite element discretization to the above discretization, we prove and numerically demonstrate the robustness of the preconditioner. Therefore, we accomplish a desirable preconditioning design goal by using the same family of preconditioners to solve the elliptic family of PDEs with varying discretizations. We also present a strategy on how to generalize the proposed preconditioner to cover high-contrast elliptic PDEs of order 2k, k>2. Moreover, we prove a fundamental qualitative property of the solution to the high-contrast biharmonic plate equation. Namely, the solution over the highly bending island becomes a linear polynomial asymptotically. The effectiveness of our preconditioner is largely due to the integration of this qualitative understanding of the underlying PDE into its construction. Copyright (C) 2010 John Wiley ; Sons, Ltd., NSFNational Science Foundation (NSF) [DMS-1016190]; Direct For Mathematical ; Physical ScienNational Science Foundation (NSF)NSF - Directorate for Mathematical ; Physical Sciences (MPS) [1016190] Funding Source: National Science Foundation, This work is supported by NSF under grant number DMS-1016190.

We investigate the well-posedness, the exponential stability, or the lack thereof, of thermoelastic systems in materials where, in contrast to classical thermoelastic models for Kirchhoff type plates, two temperatures are involved, related by an elliptic equation. The arising initial boundary value problems for different boundary conditions deal with systems of partial differential equations involving Schrödinger like equations, hyperbolic and elliptic equations, which have a different character compared to the classical one with the usual single temperature. Depending on the model -- with Fourier or with Cattaneo type heat conduction -- we obtain exponential resp. non-exponential stability, thus providing another examples where the change from Fourier's to Cattaneo's law leads to a loss of exponential stability., Peer Reviewed, Postprint (author's final draft)