Monge–Ampére simulation of fourth order PDEs in two dimensions with application to elastic–electrostatic contact problems

2017 ◽  
Vol 349 ◽  
pp. 328-350 ◽  
Author(s):  
Kelsey L. DiPietro ◽  
Alan E. Lindsay
Keyword(s):  
Fourth Order ◽  
Contact Problems ◽  
Two Dimensions ◽  
Monge Ampere

scholarly journals Robust nonconforming virtual element methods for general fourth-order problems with varying coefficients

2021 ◽  
Author(s):  
Andreas Dedner ◽  
Alice Hodson
Keyword(s):  
Fourth Order ◽  
Perturbation Parameter ◽  
Two Dimensions ◽  
Optimal Error Estimates ◽  
Projection Operators ◽  
Varying Coefficients ◽  
Virtual Element Methods ◽  
Virtual Element ◽  
Perturbation Problems ◽  
Fourth Order Problems

Abstract We present a class of nonconforming virtual element methods for general fourth-order partial differential equations in two dimensions. We develop a generic approach for constructing the necessary projection operators and virtual element spaces. Optimal error estimates in the energy norm are provided for general linear fourth-order problems with varying coefficients. We also discuss fourth-order perturbation problems and present a novel nonconforming scheme which is uniformly convergent with respect to the perturbation parameter without requiring an enlargement of the space. Numerical tests are carried out to verify the theoretical results. We conclude with a brief discussion on how our approach can easily be applied to nonlinear fourth-order problems.

The Conservative Splitting High-Order Compact Finite Difference Scheme for Two-Dimensional Schrödinger Equations

2017 ◽  
Vol 15 (01) ◽  
pp. 1750079
Author(s):  
Bo Wang ◽  
Dong Liang ◽  
Tongjun Sun
Keyword(s):  
Difference Scheme ◽  
Fourth Order ◽  
Two Dimensions ◽  
High Order ◽  
Optimal Error Estimate ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Two Dimensional ◽  
Time Step ◽  
Order Accuracy

In this paper, a new conservative and splitting fourth-order compact difference scheme is proposed and analyzed for solving two-dimensional linear Schrödinger equations. The proposed splitting high-order compact scheme in two dimensions has the excellent property that it preserves the conservations of charge and energy. We strictly prove that the scheme satisfies the charge and energy conservations and it is unconditionally stable. We also prove the optimal error estimate of fourth-order accuracy in spatial step and second-order accuracy in time step. The scheme can be easily implemented and extended to higher dimensional problems. Numerical examples are presented to confirm our theoretical results.

Anumerical algorithm for the solution of two-dimensional rough contact problems

2005 ◽  
Vol 40 (5) ◽  
pp. 463-476 ◽  
Author(s):  
M Ciavarella ◽  
G Demelio ◽  
C Murolo
Keyword(s):  
Electrical Conductance ◽  
Contact Problems ◽  
Two Dimensions ◽  
Winkler Foundation ◽  
Discrete Problem ◽  
Two Dimensional ◽  
Compliance Matrix ◽  
Linear Layer ◽  
Non Linear ◽  
Foundation Model

In this paper, a numerical algorithm is developed to solve the elastic contact problem accurately for two-dimensional rough surfaces. A first version of the method gives a full numerical solution for the discrete problem with all the details of the profile included, and the second version simulates approximately the roughness on a smaller scale with the presence of a non-linear elastic layer (as in the classical Winkler foundation model). In the literature, usually the solution of line contact is given by assuming displacements relative to a datum point, to overcome the difficulty that in two dimensions the displacements are undefined to an arbitrary constant. In the present work, the compliance matrix of the elastic half-plane is calculated starting from a self-equilibrated load distribution with periodic boundary conditions. Some examples are shown to validate the methods. Finally, the method is applied to discuss previous results by the present authors on rough contact problems defined by Weierstrass series profiles, and a discussion follows. In particular, it is found that the Winkler non-linear layer model is surprisingly useful for evaluating the electrical conductance, since (at least in the limited case of two superposed sinusoids) it does not require the wavelength and amplitude of the microscopic component of roughness to be much smaller than the macroscopic component. Some aspects of the mutual role of various components of roughness in the compliance and conductance are elucidated by means of example cases.

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