Abstract In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations (PDEs) in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are potentially free of the curse dimensionality many different applications and have proven to be so case some methods parabolic PDEs. this paper, we revi...

We propose a deep learning based method, the Deep Ritz Method, for numerically solving variational problems, particularly the ones that arise from partial differential equations. The Deep Ritz method is naturally nonlinear, naturally adaptive and has the potential to work in rather high dimensions. The framework is quite simple and fits well with the stochastic gradient descent method used in d...

We show that arbitrary convergence behavior of Ritz values is possible in the Arnoldi method and we give two parametrizations of the class of matrices with initial Arnoldi vectors that generates prescribed Ritz values (in all iterations). The second parametrization enables us to prove that any GMRES residual norm history is possible with any prescribed Ritz values (in all iterations), provided ...

in this paper free vibration analysis of two rectangular isotropic plates, which are connected to each other by two translational and rotational springs along the edges, are investigated. the equation of motion and associated boundary and continuity conditions are derived using the extended hamilton principle. to solve the eigenvalue problem, the ritz method is utilized. numerical investigation...

This article investigates the theoretical and numerical analysis as well applications of three-dimensional theory first strain gradient elasticity. The corresponding continuous discrete variational formulations are established with error estimates stemming from continuity coercivity within a Sobolev space framework. An implementation isogeometric Ritz–Galerkin method is provided open-source sof...

Modal parameters obtained from modal testing (such as modal vectors, natural frequencies, and damping ratios) have been used extensively in system identi"cation, "nite element model updating, and structural health monitoring. As an alternative to modal vectors, load-dependent Ritz vectors have been shown useful in various areas of structural dynamics such as model reduction and damage detection...

a comparison of buckling analysis of the nanoplate and nanocomposite plate with central square hole embedded in winkler foundation is presented in this article. in order to enhance mechanical properties of nanoplate with central cut out, carbon nano tubes (cnts) are applied as uniform distribution (ud) through thickness direction. in order to define the shape function of plate with square cut o...