Boundary Value Problems In Mechanics Research Articles

Brain-inspired spiking neural networks in Engineering Mechanics: a new physics-based self-learning framework for sustainable Finite Element analysis

AbstractThe present study aims to develop a sustainable framework employing brain-inspired neural networks for solving boundary value problems in Engineering Mechanics. Spiking neural networks, known as the third generation of artificial neural networks, are proposed for physics-based artificial intelligence. Accompanied by a new pseudo-explicit integration scheme based on spiking recurrent neural networks leading to a spike-based pseudo explicit integration scheme, the underlying differential equations are solved with a physics-informed strategy. We propose additionally a third-generation spike-based Legendre Memory Unit that handles large sequences. These third-generation networks can be implemented on the coming-of-age neuromorphic hardware resulting in less energy and memory consumption. The proposed framework, although implicit, is viewed as a pseudo-explicit scheme since it requires almost no or fewer online training steps to achieve a converged solution even for unseen loading sequences. The proposed framework is deployed in a Finite Element solver for plate structures undergoing cyclic loading and a Xylo-Av2 SynSense neuromorphic chip is used to assess its energy performance. An acceleration of more than 40% when compared to classical Finite Element Method simulations and the capability of online training is observed. We also see a reduction in energy consumption down to the thousandth order.

Numerical and experimental modeling of the behavior of flexible shell elements of structures

This article is devoted to computer and experimental modeling of the behavior of flexible elements of shell structures, and the development of effective algorithms for solving emerging nonlinear boundary value problems of their calculation and optimization of parameters. At the same time, the probability of the results obtained using different approaches to the construction of a nonlinear theory is established. Their comparative analysis, error estimation, which in this case is given by calculation according to linear and corresponding nonlinear theories, is carried out. The results of calculated data and experimental studies of the behavior of real structural elements are compared.
 The results of a comparative analysis of the application of two mathematical models of deformation of flexible shell elements, obtained by direct integration of boundary value problems of shell mechanics, by the finite element method and experimental research, are presented.
 The problems of axisymmetric nonlinear bending of thin ring plates, corrugated membranes of a sinusoidal profile and bellows as a shell of rotation with a complex meridian shape are considered.
 Using the necessary optimality conditions of the principle of maximum L. S. Pontryagin obtained the results of the optimal design of a flexible corrugated membrane with a sinusoidal profile of the highest sensitivity. The results are presented in the form of tables, photos and graphs.

Spiking recurrent neural networks for neuromorphic computing in nonlinear structural mechanics

The present study aims to introduce an AI algorithm suitable for neuromorphic computing to solve Boundary Value Problems in Engineering Mechanics. Following the trend of sustainable Artificial Intelligence (AI), Spiking Neural Networks (SNNs), known as the third generation of neural networks, are proposed for developing surrogate models for mechanical tasks. SNNs are inherently recurrent and communicate through sparse signals consuming only a fraction of the energy needed by artificial neural networks (ANNs) of the second generation such as current deep learning methods. To take advantage of the energy-efficient spike models, an autoencoding strategy is proposed to convert the real-valued data at a certain time step to its binary spike representation which allows the model to be used for regression tasks. Further, we propose a hybrid model consisting of the spiking variant of the Legendre Memory Unit (LMU), spiking recurrent cells, and classical dense transformations to compute the nonlinear (physically and geometrically) response of shock wave-loaded plate elements. The inclusion of spikes enables the proposed model to be deployed on neuromorphic hardware, such as the Loihi chip. Thus, for a series of both training and validation experiments, a detailed comparison of the proposed hybrid model and its second-generation counterpart in terms of energy consumption and accuracy is presented.

BINN: A deep learning approach for computational mechanics problems based on boundary integral equations

We proposed the boundary-integral type neural networks (BINN) for the boundary value problems in computational mechanics. The boundary integral equations are employed to transfer all the unknowns to the boundary, then the unknowns are approximated using neural networks and solved through a training process. The loss function is chosen as the residuals of the boundary integral equations. Regularization techniques are adopted to efficiently evaluate the weakly singular and Cauchy principal integrals in boundary integral equations. Potential problems and elastostatic problems are mainly concerned in this article as a demonstration. The proposed method has several outstanding advantages: First, the dimensions of the original problem are reduced by one, thus the degrees of freedom are greatly reduced. Second, the proposed method does not require any extra treatment to introduce the boundary conditions, since they are naturally considered through the boundary integral equations. Therefore, the method is suitable for complex geometries. Third, BINN is suitable for problems on the infinite or semi-infinite domains. Moreover, BINN can easily handle heterogeneous problems with a single neural network without domain decomposition.

Decoupling and quasi-linearization methods for boundary value problems in relative orbital mechanics

Decoupling and quasi-linearization methods for boundary value problems in relative orbital mechanics

A model of thermal creep and annealing in finite domains based on coupled dislocation climb and vacancy diffusion

We develop a framework to investigate thermal creep and annealing in finite domains, where the climb motion of discrete dislocations is coupled to the diffusion of a continuum vacancy field. The model is first formulated in a continuum finite-deformation setting. All governing equations and boundary conditions are obtained from a unified irreversible thermodynamics principle. The resulting model couples a mechanical boundary value problem (BVP), a vacancy diffusion BVP, and the climb and glide motion of the discrete dislocation network within the crystal. The framework is then linearized for implementation in three-dimensional (3D) discrete dislocation dynamics (DDD) simulations for arbitrary anisotropic crystals. A solution scheme is developed based on the superposition principle, which is imposed weakly on the dislocation network to obtain a Galerkin solution for the nodal climb velocities. The framework includes diffusional (Nabarro–Herring) creep deformation as well as dislocation creep by climb-assisted-glide. The method is applied to simulate the annealing of vacancy loops in Al, with good agreement to experimental measurements by Silcox and Hirsch (1959). We further consider the effects of annealing under stress, and of the proximity of the vacancy loops to loaded and free boundaries Simulations in polycrystalline materials are carried out to highlight the effects of the grain size on dislocation climb and vacancy loop annealing. The method is also applied to estimate the creep rate due to climb-assisted glide of jogged-screw dislocations in γ-TiAl, and results are compared to experiments by Viswanathan et al. (1999). Finally, we discuss the effects of uniaxial and hydrostatic stresses on the two diffusive deformation pathways of the material, namely Nabarro–Herring creep and dislocation climb.

NN-EUCLID: Deep-learning hyperelasticity without stress data

We propose a new approach for unsupervised learning of hyperelastic constitutive laws with physics-consistent deep neural networks. In contrast to supervised learning, which assumes the availability of stress–strain pairs, the approach only uses realistically measurable full-field displacement and global reaction force data, thus it lies within the scope of our recent framework for Efficient Unsupervised Constitutive Law Identification and Discovery (EUCLID) and we denote it as NN-EUCLID. The absence of stress labels is compensated for by leveraging a physics-motivated loss function based on the conservation of linear momentum to guide the learning process. The constitutive model is based on input-convex neural networks, which are capable of learning a function that is convex with respect to its inputs. By employing a specially designed neural network architecture, multiple physical and thermodynamic constraints for hyperelastic constitutive laws, such as material frame indifference, material stability, and stress-free reference configuration are automatically satisfied. We demonstrate the ability of the approach to accurately learn several hidden isotropic and anisotropic hyperelastic constitutive laws – including e.g., Mooney–Rivlin, Arruda–Boyce, Ogden, and Holzapfel models – without using stress data. For anisotropic hyperelasticity, the unknown anisotropic fiber directions are automatically discovered jointly with the constitutive model. The neural network-based constitutive models show good generalization capability beyond the strain states observed during training and are readily deployable in a general finite element framework for simulating complex mechanical boundary value problems with good accuracy.

ITERATIVE ALGORITHMS FOR SOLVING A PLANE PROBLEM FOR AN ENVIRONMENT WITH SIGNIFICANT MANIFESTATION OF INTERNAL FRICTION

The purpose of research is to develop an algorithm for solving of a flat nonlinear problem of mechanics of discrete environment, which laws of deformation of take into account the effect of the internal friction. Described the explanation of the iterative algorithm for the numerical solution of the plane physically nonlinear boundary value problems of mechanics of discrete environment. Feature of the problem is the consideration of the effect of internal Coulomb friction on the deformation of the environment. When solving the problem in displacements for finite simplex elements, the continuity conditions are always satisfied at the nodes and at the faces of the element. Solving the system of canonical linear equations of the displacement method ensures the fulfillment of the equilibrium conditions in each node of the discrete computational domain. Therefore, the calculation procedure is organized in such a way that the obtained solutions also comply with the laws of deformation of the material. Known iterative methods of variable stiffness, initial stresses or initial deformations, which differ only in the method of obtaining solutions, can be used for this purpose. The analysis showed that the most effective method for solving the formulated physically nonlinear problem is the variable stiffness, on the basis of which the iterative algorithm was developed. Described in the article iterative algorithm for solving of a flat boundary value problems of mechanics of a discrete environment allows to take into account the influence on the process of deformation internal friction. It can also be used to solve problems in the mechanics of solid deformable body with a most influence of internal friction, thermo-elasticity problems, etc.

Fluid Flow Development in a Pipe as a Demonstration of a Sequential Change in Its Rheological Properties

The sequence of the process of changing the velocity profiles and the laws of resistance during the flow of a fluid in a pipe is considered. With the increasing of the Reynolds number, we obtain the transition of the flow regime from laminar to turbulent. In the presence of small additives of polymers, when the Toms effect is observed in the fluid flow, the turbulent regime changes with a further increase in the Reynolds number to another regime, the rheology of which leads to laminar velocity profiles and corresponding resistance laws. Then, with an increase in the Reynolds number for polymer solutions, the limiting Virk flow regime with its own rheology is reached. All the mentioned flow regimes and all types of rheology can be described using one rheological relation, which is a power-law generalization of Newton’s formula, by changing the values of the power value in this ratio upon reaching the corresponding critical Reynolds numbers. This generalization can be extended to the spatial case of flow and the rheological relation can be represented in tensor form with a further system of differential equations for a fluid flow with an arbitrary rheology. After that, boundary value problems in fluid mechanics can be solved for any fluid flow regime.

Vectorized numerical algorithms for the solution of continuum mechanics problems

The aim of the work is to study the possibilities provided by new information technologies, object-oriented programming tools and modern operating systems for solving boundary value problems of continuum mechanics described by partial differential equations. To discretize the basic equations, we applied the method of finite differences and finite volumes, which are widely used to solve problems in the mechanics of liquids and gases. The paper considers the implementation of the finite difference methods and the finite volume method with vectorized grid structures, including access to the inner and boundary cells of the grid, as well as the features of the implementation of algorithms at singular points of the computational domain. To solve boundary value problems described by partial differential equations, we developed an approach to the construction of vectorized algorithms and considered the features of their software implementation in the MATLAB package. Vectorization in such tasks, excluding nested loops, is ensured by appropriate data organization and the use of vectorized operations. On the one hand, the developed algorithms widely use MATLAB functions designed for processing vectors and sparse matrices, and on the other hand, they are distinguished by high efficiency and computation speed, comparable to those of programs written in C/C++. The main results imply the numerical solution of a number of problems in continuum mechanics associated with the calculation of stresses in a separate body and the calculation of the field of velocity and temperature in the flow of a viscous incompressible fluid. The features of discretization of the basic equations and the implementation of the corresponding finite-difference and finite-volume algorithms are shown. The use of the MATLAB system opens up new possibilities for the formalization and implementation of finite-difference and finite-volume methods for the numerical solution of boundary value problems in continuum mechanics. Despite the fact that the capabilities of the developed algorithms are illustrated by the example of solving fairly simple problems, they admit a relatively simple generalization to more complex problems, for example, solving the Euler equations and Navier–Stokes equations. As part of the work, computational modules were prepared with user programming tools that expand the capabilities of the package and are focused on solving problems in continuum mechanics.

A computationally efficient thermo-mechanical model for wire arc additive manufacturing

Residual stresses and distortions are major obstacles against the more widespread application of wire arc additive manufacturing. Since the steep temperature gradients due to a moving localised heat source are inevitable in this process, accurate prediction of the thermally induced residual stresses and distortions is of paramount importance. In the present study, a computationally efficient thermo-mechanical model based on a semi-analytical thermal approach incorporating Goldak heat sources is developed for the process modelling of wire arc additive manufacturing. The semi-analytical thermal model makes use of the superposition principle, and thereby decomposes the temperature field into an analytical temperature field to account for the heat sources in a semi-infinite space and a complementary temperature field to account for the boundary conditions. Since the steep temperature gradients are captured by the analytical solution, a coarse spatial discretisation can be used for the numerical solution of the complementary Tˆ field. Thermal evolution is coupled to an elasto-plastic mechanical boundary value problem that computes the thermal stresses and distortions. The accuracy of the proposed model is evaluated extensively by comparing the thermal and mechanical predictions with the corresponding experimental measurements as well as the simulation results obtained by a non-linear transient model from the literature. A thin wall structure with a length of 500 mm and consisting of 4 layers is modelled. The peak normal stress along the deposition direction can be predicted with less than 10% error. Furthermore, the simulations show that the part distortions are very sensitive to the boundary conditions.

Analysis of Timoshenko–Ehrenfest beam problems using the Theory of Functional Connections

The numerical solution of boundary value problems in solid mechanics is dominated by the Finite Element Method (FEM). The present study provides an alternative numerical approach using the Theory of Functional Connections (TFC) for solving problems of this type. Here TFC is used to solve multiple beam problems using the Timoshenko–Ehrenfest beam theory which accounts for transverse shear strain. The results obtained are then compared to those obtained with both the FEM and the analytical approaches. For cases where the analytical solution is continuous and smooth, TFC out performs FEM both in terms of the L2 norm of the residuals and the solution time. When the analytical solution is continuous and smooth, TFC results in an L2 norm on the order of machine-error level precision. However, when the higher order derivatives of the analytical solution are discontinuous (i.e. when the applied load is discontinuous), TFC under performs compared to FEM. This is due to TFC approximating the solution by continuous functions when the true solution is piece-wise continuous. In terms of computational expense, TFC consistently solved the differential equations faster than FEM across each type of problem analyzed in this paper.

Numerical analysis of an MHD flow in fuzzy environment in presence of induced magnetic field

AbstractDue to the uncertain behavior of real‐life problems, fuzzification is one of the most suitable ways to find the interval at which the problems show their accurate results. Various imprecise parameters and conditions make the fluid flow problems more uncertain. Here, we try to solve a boundary value problem of fluid mechanics numerically using fuzzy conditions. The fuzzified governing partial differential equations along with the fuzzified boundary conditions are solved by finite difference method using Python (an object‐oriented programming language). The values of the parameters and the boundary conditions are taken as triangular fuzzy numbers. ‐cut technique is used to find the results and they are presented graphically for various values of and the involved parameters.

Green tensor and regular solutions of equations of rods thermodynamics and their properties

1. Alekseyeva L.A., Dadayeva A.N., Zhanbyrbaev N.B., 1999, The method of boundary integral equations in boundary value problems of unconnected thermoelastodynamics, Applied Mathematics and Mechanics, 63, 5, 803-808. CrossRef Google Scholar

ВПЛИВ ГЕОМЕТРИЧНОЇ ФОРМИ ПАЛІ НА ЇЇ ОПІР ЗА МГЕ

Vinnytsia National Technical University The estimated cost of construction of foundations reaches about 40% of the total estimated cost of the building. In the industrial and civil construction of Ukraine due to its geological conditions it is advisable to use pyramidal piles and the peculiarities of their interaction with the foundations and theoretical calculation methods are insufficiently studied. The lack of reliable methods for estimating the bearing capacity of pyramidal piles leads to an increase in inventory ratios and imperfect decisions. Therefore, the topic of VAT research of rational pyramidal piles is relevant. Numerical studies on the IHE to determine the VAT and bearing capacity of pyramidal piles, which are characterized by load resistance with equal subsidence. Numerical studies of the pyramidal pile were performed on the numerical IHE. MGE - a modern numerical method for solving boundary value problems of structural mechanics, has attracted the attention of researchers mainly by the fact that the continuous environment is divided into a number of elements that can be considered as separate parts The analysis of results of numerical researches is carried out, the correlation with experimental data which are restrained by direct measurement of pressure in a soil basis by mezdoses is carried out.

Modeling the deformation process of a plate with a stress concentrator taking into account the postcritical stage of material deformation

Postcritical deformation of a material is a process which is characterized by a decrease of stress during growing deformations as a result of accumulation of structural damage. The design becomes unable to withstand the external load only when zones with weakened connections are developed enough. Evolution of postcritical deformation zones can occur with an increase of the external load applied to the construction. It means that taking into account the softening of the material allows determining the strength and deformation reserves of constructions more accurately. The mathematical formulation of the boundary value problem of supercritical deformation mechanics is given in the paper. The features of the experimental study of the postcritical stage of material deformation are listed. Strain curves of various steels with a long section of softening are obtained. Numerical solutions for the problems of deforming a thin plate with stress concentrators of different geometries under kinematic loading are obtained. Piecewise linear approximations and real strain curves of steel 20 and steel 40Cr4 obtained experimentally are considered. The evolution of zones of postcritical deformation in the material is considered. The correspondence between the value of the decline modulus and the nature of the evolution of the softening zones is determined. A stress plot is constructed that reflects how the complete material deformation diagram is realized near the concentrator. The calculated loading diagrams are constructed. It is noted that even after the appearance of softening zones, an increase in external load is possible. The strength and deformation resources of structures are determined, and the influence of the geometry of the stress concentrator on their values is considered. It is noted that the consideration of softening in modeling the behavior of structures with stress concentrators is appropriate.

The Block-Element Method in Expansion of the Solutions of Complex Boundary-Value Problems in Mechanics

An exact solution in the first quadrant of a plane boundary-value problem of the second kind for the dynamic Lame elasticity equations is constructed, probably for the first time, and is expanded in terms of solutions of boundary-value problems for the Helmholtz equation. These solutions are presented in the form of packed block elements.

Approximate analytical and numerical solutions of a nonlinear boundary value problem in fluid mechanics based on symmetry reduction

This work presents applications of the L i e symmetry method to a boundary value problem (BVP) of the governing equations in fluid mechanics. The Wu differential characteristic set algorithm is used to obtain a group of multi-parameter symmetries for the BVP, and the L i e symmetry method is adopted to simplify the BVP into some initial value problems of reduced diffusion equations. Approximate analytical and numerical solutions are then obtained by utilizing the homotopy perturbation method and the Runge–Kutta method respectively, and compared to exhibit the convergence of approximate analytical solutions. The formulation developed here could be applied to studies of other nonlinear BVPs in fluid mechanics.

The method of single expression (MSE) as a prospective modeling tool for boundary value problems: an extension from nano-optics to quantum mechanics

Mathematical description of the wave phenomena in nano-optics and quantum mechanics is similar and requires wavelength-scale analysis of wave interaction with nano-layers in optics and micro-particle interaction with potential barriers or wells in quantum mechanics. Traditionally, when dealing with boundary problems in nano-optics and quantum mechanics, the same fundamental approach of counter-propagating waves is often being used, when general solutions of the wave equations are presented as a sum of counter-propagating waves. This type of solution presentation relies on the superposition principle restricting correct description of strong intensity-dependent nonlinear wave-matter interaction. The non-traditional method of single expression (MSE) does not exploit the superposition principle, but rather uses resulting field representation and backward-propagation algorithm allowing to obtain correct steady-state solutions of boundary value problems without approximations and at any value of wave intensity by taking into account correctly intensity-dependent nonlinearity, loss or gain in a medium. In the present work a detailed description of the MSE approach extended for one dimensional quantum mechanical boundary value problems is presented. Results of numerical simulations by the MSE of electron tunneling through rectangular single and double potential barriers are presented and discussed.

The Interfacial Nature of Mechano-Chemo-Electrical Potential and Its Application to Processing and Failure Prediction in Solid State Lithium Ion Batteries

The incorporation of mechanical stresses into descriptions of the electrochemical potential is of fundamental significance in understanding the processing and failure mechanisms in solid state lithium ion batteries. The chemical potential is defined by (i) the nature of the species, e.g. lithium atoms, or ions, and (ii) by the site or the phase where the species reside. The second point draws a distinction between the influence of mechanical tractions applied to a surface or an interface – e.g. grain boundaries – and the volumetric strain energy obtained from solutions to the usual boundary value problems in mechanics. This distinction is great, and not sufficiently recognized in the literature. In this presentation the interfacial mechano-chemo-electrical potential will be described and then applied to three problems in the processing and failure prediction of solid state lithium ion batteries made with Li-metal anodes.