Generalized variational principles with relaxed continuity requirements for certain nonlinear problems, with an application to nonlinear elasticity

In the article we obtained sufficient conditions of the existence of the nonlinear Noetherian boundary value problem solution for the system of differential-algebraic equations which are widely used in mechanics, economics, electrical engineering, and control theory. We studied the case of the nondegenerate system of differential algebraic equations, namely: the differential algebraic system that is solvable relatively to the derivative. In this case, the nonlinear system of differential algebraic equations is reduced to the system of ordinary differential equations with an arbitrary continuous function. The studied nonlinear differential-algebraic boundary-value problem in the article generalizes the numerous statements of the non-linear non-Gath boundary value problems considered in the monographs of А.М. Samoilenko, E.A. Grebenikov, Yu.A. Ryabov, A.A. Boichuk and S.M. Chuiko, and the obtained results can be carried over matrix boundary value problems for differential-algebraic systems. The obtained results in the article of the study of differential-algebraic boundary value problems, in contrast to the works of S. Kempbell, V.F. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and A.A. Boychuk, do not involve the use of the central canonical form, as well as perfect pairs and triples of matrices. To construct solutions of the considered boundary value problem, we proposed the iterative scheme using the method of simple iterations. The proposed solvability conditions and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem, were illustrated with an example. To assess the accuracy of the found approximations to the solution of the nonlinear differential-algebraic boundary value problem, we found the residuals of the obtained approximations in the original equation. We also note that obtained approximations to the solution of the nonlinear differential-algebraic boundary value problem exactly satisfy the boundary condition.

Optimum design of the shape of the initial blank is an important task before sheet metal forming processes. It reduces production costs, material waste and improves the quality of the product significantly. In the present work, the blank shape optimization problem is considered based on the 3D space target contours and moreover the sheet metal can have an internal boundary. In other words, the shape optimization process can be applied on the internal boundary, external boundary or both. To the best knowledge of the authors, there are no published works in this subject. Using the iterative simulation-based optimization process, a special shape error index and also updating algorithm were proposed to modify the blank geometry in each iteration until capturing the optimum shape. The sheet forming process is highly nonlinear in nature due to plastic behaviors, large deformations and frictional contact surfaces. Therefore, the updating formula should be robust enough to have less sensitivity with respect to the initial guess. To evaluate the proposed updating formula and its robustness, some numerical examples were considered and the effects of different tools geometries, 2D and 3D target contours, internal and external boundaries and different initial guesses were examined.

A new technique for solving the large system of linear algebraic equations associated with implicit differencing of multidimensio nal partial differential equations is presented. The coefficient matrix of the equations is factored, and then approximations to certain terms in the matrix are obtained from series expansions. The resulting system of equations is solved easily. The method is developed and demonstrated using a simple representative two-dimensional equation. Very good results are obtained when one direction is dominant. MPLICIT finite-difference schemes for the solution of multidimensional partial differential equations are usually stable and therefore applicable to a large class of problems. However, they are difficult to implement and may require an excessive amount of computer storage and time. The long computing time arises from the need to solve the large system of linear algebraic equations that result from the differencing. The computing time can be reduced significantly by approximating the coefficient matrix of the linear equations with a matrix that produces a system of equations that are relatively easy to solve. Among such methods are the alternating direction method (ADI)l used by Beam and Warming2 and Stone's strongly implicit method,3 which has been tested by Linetal.4 In this paper, a new technique for solving the large system of linear algebraic equations associated with implict differencing of multidimensional partial differential equations is presented. This method, called the pseudo-elimination method (PE), is shown to be faster than Stone's method for certain problems. The method is directly applicable to linear and linearized nonlinear systems of parabolic or elliptic partial differential equations. In order to discuss the method, a simple linear partial differential equation will be used; however, it should be kept in mind that the PE method is applicable to much more complicated problems. The question of whether the method will work when applied to difficult problems is not addressed. The scope of this paper is limited to presenting the method and illustrating, via a simple problem, that the method has some merit and deserves further study.

Chapter 5 - Life Is Really Nonlinear

A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed.

Design optimization of dynamic flexible multibody systems using the discrete adjoint variable method

The manuscript investigates the numerical approximation of the fractional mathematical model of the financial crime population dynamics by the Gegenbauer wavelet collocation method. The study aims to enhance the accuracy and efficiency of solving the underlying differential equations that describe these phenomena by utilizing the proposed technique. The financial crime model is a nonlinear coupled system of ordinary differential equations. Using the Gegenbauer wavelets, the novel operational matrices of integration are created. A nonlinear system of ordinary differential equations are transformed into a system of algebraic equations using the characteristics of the Gegenbauer wavelet expansions and the operational matrix of integration, which speeds up processing. Then, this system of algebraic equations is solved using the Newton‐iterative technique to find the unknown Gegenbauer coefficients that help to obtain the approximate solution for the system. A numerical illustration is presented to show the efficacy and precision of the approach. The numerical results obtained from the projected approach are compared with the existing methods, such as NDSolve and Runge Kutta methods. These results show that the projected scheme is simple, reliable, and resilient. The findings suggest that this approach can be a powerful tool for researchers and practitioners in the financial sector, aiding in developing crime prevention and intervention strategies. The study concludes with suggestions for future research directions.

Rational general solutions of systems of first-order algebraic partial differential equations

The dynamic models of processes in transport logistics systems are considered. In the literature, such complex systems as the military transport logistics system or the combat vehicle support system are often modeled as a set of typical system states. These states are interconnected by a large number of transitions of a given intensity, which are carried out with given probabilities. Graphically, this is represented using the so-called graph of states and transitions, and the probabilities of the system being in a particular state are the subject of research in such a graph. The methods available in the literature for studying the dynamic characteristics of state graphs and transitions are analyzed. A description of the changes in probabilities as a function of time is made using systems of differential equations, usually linear. Based on practical requirements, approximate solutions to such systems are usually sought. One of the approximate methods is the decomposition method, in which, instead of a system of coupled equations, a set of independent equations is considered, the solution of which is not difficult. The results of the solution have an accuracy satisfactory from the point of view of practical use. The assumptions based on which the decomposition method can be used are analyzed. It is shown that the accuracy of the obtained results substantially depends on the given initial conditions and should increase over time, when this dependence weakens. A method is proposed for the exact solution of a system of differential equations, free of any assumptions. The use of operational calculus is substantiated, which reduces the solution of a system of linear differential equations to the solution of a system of linear algebraic equations for unknown images of the sought-for Laplace functions. The method is used to describe the process of technical support for the restoration of the transport flow of military logistics. The boundaries of the possibility of applying the results of a simpler approximate solution are established.

In this paper, we construct an optimization problem to solve a system of algebraic equations over a finite field. Given a system of equations over finite field, we derive a system of equations over the real numbers from the given system. We construct a model of an unconstrained optimization problem by using a penalty function. We also show that an optimal solution to the unconstrained optimization problem is the solution of the original system of algebraic equations over a finite field.

There are no strict rules for the numerical treatment of partial differential equations. We concentrate here on linear partial differential equations. As an example of elliptic partial differential equations the two-dimensional Poisson equation with Dirichlet boundary conditions is investigated. The application of the finite difference approximation transforms this equation into a set of algebraic equations which can be solved iteratively. The time dependent heat equation is an example of parabolic partial differential equations. The one-dimensional Dirichlet boundary value problem is transformed using the implicit and explicit Euler method into an inhomogeneous system of linear algebraic equations. Stability problems of the explicit Euler method are discussed and demonstrated numerically. The wave equation is a model for hyperbolic partial differential equations. It can be transformed into a system of algebraic equations using, for instance, the explicit Euler method. Again, stability problems connected with this method are discussed and demonstrated numerically. Finally, the quantum mechanical tunneling effect is studied numerically by solving the time dependent Schrodinger equation by means of a Crank-Nicholson scheme which transforms the Schrodinger equation into a system of linear algebraic equations with tridiagonal matrix.

In this paper we present a technique to study the existence of rational solutions for systems of differential equations — for an ordinary differential equation, in particular. The method is relatively straightforward; it is based on a rationality characterisation that involves matrix Pade approximants. It is important to note that, when the solution is rational, we use formal power series “without taking into account” their circle of convergence; at the end of this paper we justify this. We expound the theory for systems of linear first-order ordinary differential equations in the general case. However, the main ideas are applied in numerical resolution of partial differential equations.

In powder metallurgy, a common process is hot isostatic pressing (HIP). This process is characterized by the densification of powders using a liquid or a gas. Due to the indirect heating, such processes usually take several hours. A new and innovative production process for the sintering of powders is the Field Assisted Sintering Technology (FAST). Here, the powder is filled into a graphite die and compacted by a punch. At the same time, the powder and the graphite tool system are heated volumetrically by an electric current that induces Joule heating. With this technology at hand, it is possible to ensure faster heating and shorter cooling periods, leading to shorter production times and improved material properties. The shorter production times are also beneficial for the microstructure, since grain coarsening due to large dwell times can be avoided. A major challenge of this new technology is to control the various process parameters in order to determine the desired material properties. In order to get a deeper insight into the process by means of simulations, a constitutive model is derived for the highly instationary, non-linear thermo-mechanical powder consolidation process on the basis of principal experiments. Additionally, the graphite tool system is modeled for the simulation of the entire process. The parameters of the developed model are identified with the concept of material parameter identification. The finite element method proved to be a powerful tool for the numerical simulation of physical problems. For the numerical simulation of the sintering process, the underlying thermo-electro-mechanically coupled problem has to be solved together with temperaturedependent graphite material properties, the temperature and the relative density-dependent material properties for the powder material. This time-dependent non-linear multi-field problem is interpreted as a system of differential algebraic equations and its solution procedure by means of finite elements is explained. The final stress algorithm is implemented into the commercial finite element program Abaqus, and a fully coupled monolithic simulation is performed in order to simulate a real FAST-process and to compare the temperature and stress distribution with experiments. Moreover, it is possible to make predictions concerning the powder densification and the temperature distribution, which leads to a better insight into the process. As a result, the numerical model for the simulation of the thermal, electrical and mechanical problems of sintering can be seen as a suitable tool to optimize the sintering process.

Applications of the Lanczos method

One proves theorems regarding the estimates of the solutions of systems of linear algebraic equations and inequalities. On the basis of these theorems one suggests a method for estimating the norm of the inverse matrix of a system of difference equations which approximates a boundary-value problem for an integrodifferential equation. The method allows us to eliminate the restrictions which are usually imposed on the coefficients of the integrodifferential equation in order to ensure the diagonal dominance in the system of the difference equations. One considers an application to nonlinear problems.