The so-called Ritz–Galerkin method is one of the most fundamental tools of modern computing. Its origins lie in Hilbert’s “direct” approach to the variational calculus of Euler– Lagrange and in the thesis of Walther Ritz, who died 100 years ago at the age of 31 after a long battle with tuberculosis. The thesis was submitted in 1902 in Göttingen, during a period of dramatic developments in physi...

Rayleigh–Ritz eigenvalue estimates for Hermitian matrices obey Cauchy interlacing, which has helpful implications for theory, applications, and algorithms. In contrast, few results about the Ritz values of non-Hermitian matrices are known, beyond their containment within the numerical range. To show that such Ritz values enjoy considerable structure, we establish regions within the numerical ra...

In this paper, considering an adhesively bonded composite single-lap joint, a novel approach is presented to predict the peel and shear stress distributions of the adhesive layer for an ASTM standard test sample. In current method, the equilibrium equations are derived using the energy method and based on the Timoshenko’s beam theory. Two solution procedures then are discussed, one of which rep...

In this paper, we study the deep Ritz method for solving linear elasticity equation from a numerical analysis perspective. A modified formulation using $H^{1/2}(\Gamma_D)$ norm is introduced and analyzed in order to deal with (essential) Dirichlet boundary condition. We show that resulting provides best approximation among set of neural network (DNN) functions respect ``energy'' norm. Furthermo...

The objective of this paper is to present a comparative study of fitted-mesh finite difference method, Ritz-Galerkin finite element method and B-spline collocation method for a two-parameter singularly perturbed boundary value problems. Due to the small parameters ε and μ, the boundary layers arise. We have taken a piecewise-uniform fittedmesh to resolve the boundary layers and shown that fitte...

This paper describes an experimental study on the use of Ritz vectors for damage detection of a grid-type bridge model. A new procedure to extract Ritz vectors from experimental modal analysis is proposed and demonstrated using the test structure. Using appropriate load patterns, Ritz vectors can be made more sensitive to damage than modal vectors. The results indicate that the use of load-depe...