A robust Moore–Penrose pseudoinverse-based static finite-element solver for simulating non-local fracture in solids

Solvability of a system of linear equations—an approach based on the generalized inverses determined by the Penrose equations ★

“…Visibly, when N = 0 and M = 0, the system (17) reduces to a single equation AX = C corresponding to p systems of linear equations with the same coefficient matrix A. Replacing now matrices X and C with column vectors x and c, respectively, shows that (18) takes the form (15), whereas (19) reduces to (16).…”

“…The expression (16) for the general solution of the equation (14) indicates that each and every solution can be obtained by varying the vector z. In particular, with z = 0 the solution takes the form x = A − c for arbitrary A − ä A{1}.…”

“…Despite of not so common knowledge of the Moore-Penrose inverse within physics community, relevance of the notion for the research carried out in physics is reflected by a steady stream of physics papers published annually, which refer to the inverse. In order to update the list of references provided in [3] with the most recent articles taking advantage of the Moore-Penrose inverse, let us mention that: in [16] the inverse of a stiffness matrix was used to solve the characteristic finite-element equations in considerations dealing with modeling of elastic deformation and non-local brittle fracture of solids, in [17] the inverse of a correlation matrix was used in a study of correlated pulsar-pairs, in [18] the inverse of operator matrices in a Hilbert space framework was used to derive solutions of constrained linear equations using variational principles in considerations of a class of problems (called Z-problems) that arise in effective media theory, in [19] the inverse was used to derive numerical solutions to particular nonlinear ordinary differential equations and differential algebraic equations involved in neural networks analysis, in [20] the inverse of a key matrix was used to derive a novel image encryption algorithm, in [21] the inverse was used to construct an image fusion algorithm based on a matrix factorization, which aims at optimizing computational workload, in [22] the inverse was used to solve a set of linear equations resulting from a novel variant of the multiconfiguration perturbation theory proposed in order to ensure its pivot invariance, and in [23] the inverse was used to specify new mode time coefficient (equivalent to the best-fit solution in the least squares sense) for dynamic mode decomposition utilized to extract dominant modes of unsteady flow fields. Additionally, in [24] the Moore-Penrose inverse was used in comparison of alternative models of velocity and acceleration level inverse kinematic for redundant manipulators, with the inverse of a Jacobian matrix involved in each model; orthogonal projectors onto the fundamental subspaces associated with the Jacobian matrix were utilized in the analysis as well.…”

A Review on Nonlocal Theories in Fatigue Assessment of Solids

“…A nonlocal fracture criterion introduced by Khodabakhshi et al [ 104 ] in 2016 was developed by the same authors [ 105 ] in 2019 and more recently by Alebrahim et al [ 106 ] and Shin et al [ 107 , 108 ] to investigate the fracture of quasi-brittle materials using a graph-based finite element technique. The main idea was that a network link would fail if the weighted average nonlocal strain, derived using Equation (6), exceeded a critical value, ; i.e., where is the unit vector along the edge of each element in the graph-based finite element modeling.…”

Application of Machine Learning in Fracture Analysis of Edge Crack Semi-Infinite Elastic Plate

“…Numerical simulation, on the other hand, has become an essential tool in engineering methodologies, offering an efficient means for investigating complex systems. The Finite Element Analysis (FEA) approach has gained significant acceptance in this context for investigating fracture behavior, aided by software tools such as ABAQUS CAE and ANSYS Workbench to perform extensive numerical simulations of fracture propagation in complex solid structures, resulting in a rich dataset that incorporates experimental data [12][13][14]. The application of machine learning (ML) techniques in fracture analysis has influenced the area of material engineering and structural mechanics in recent years [15,16].…”