A Review of Mathematical Models of Elasticity Theory Based on the Methods of Iterative Factorizations and Fictitious Components

Abstract

This review analyzes articles on mathematical modeling of elasticity theory using iterative factorizations and fictitious components. To carry out this study, various methods are developed, for example, an approximate analytical method of iterative factorizations for calculating the displacements of a rectangular plate, and a modified method of fictitious components for calculating the continuous displacement of plates. The performed calculations confirm the effectiveness of these methods. Descriptions of solutions to problems of elasticity theory and possible applications of the considered mathematical models and methods are given. An overview of the methods used to solve these problems is given. Particular attention is paid to problematic issues that arise in solving these problems. Techniques used to reduce complex problems to the solution of simple problems are given, for example, the lowering of the order of differential equations and the reduction of solutions in complex domains to solutions in a simple domain. For the first approach, iterative factorization methods are often used, and for the second, methods of the fictitious component type are often used. The main presentation in this review is focused on the approximate solution of elliptic boundary value problems. The works considered in the review raise questions about the development of methods in research on fictitious domains, fictitious components, and iterative factorizations.

1. Introduction

Various physical phenomena in nature and technology, in areas such as hydrodynamics, mechanics, heat engineering, and electrical engineering, can be described by mathematical models derived from partial differential equations. or instance, stationary or steady processes are often modeled using elliptic equations of the second and fourth orders. The Poisson equation models, for example, describe the movement of membranes under the influence of applied pressure.

The numerical solution of the biharmonic equation and similar ones, even in a square region, under the main boundary conditions is not sufficiently mastered, and there are difficulties in solving such problems.

This review is devoted to solving problems of the following mathematical models.

• − The mathematical model of the displacement of the points of a rectangular membrane on an elastic base, under the action of pressure, fixed on two adjacent sides:

$$-\Delta u+\kappa u=f \mathrm{in} \Omega , u|{}_{{\Gamma }_1}=0 \frac{\partial u}{\partial n}|{}_{{\Gamma }_2}=0$$

(1)

where Ω is the rectangular area $\Omega =(0;b_1)\times (0;b_2),$ with a border $\hspace{0.17em}\partial \Omega =\hspace{0.17em}\overline{s},\hspace{0.17em}\hspace{0.17em}s={\Gamma }_1\cup {\Gamma }_2,$ $\hspace{0.17em}\partial \Omega =\hspace{0.17em}\overline{s},\hspace{0.17em}\hspace{0.17em}s={\Gamma }_1\cup {\Gamma }_2,$ ${\Gamma }_1=\left\{b_1\right\}\times (0;b_2)\cup (0;b_1)\times \left\{b_2\right\},\hspace{0.17em}\hspace{0.17em}{\Gamma }_2=\left\{0\right\}\times (0;b_2)\cup (0;b_1)\times \left\{0\right\},$ $n$ is outward normal to $\partial \Omega$, $P$ is pressure, $K\ge 0$ is the coefficient of rigidity of the elastic foundation, $T$ is membrane tension factor, $u$ is the desired displacement, $\kappa =K/T$, and $f=P/T$.

• − The mathematical model of the displacement of the points of a rectangular plate on an elastic foundation under the action of pressures, where on two adjacent sides there are homogeneous conditions of hinged support, and on the other two sides there are homogeneous symmetry conditions:

$${\Delta }^{2}u+au=f \mathrm{in} \Omega , u|{}_{{\Gamma }_1}=\Delta u|{}_{{\Gamma }_1}=0, \frac{\partial u}{\partial \overrightarrow{n}}|{}_{{\Gamma }_2}=\frac{\partial \Delta u}{\partial n}|{}_{{\Gamma }_2}=0$$

(2)

where Ω is the rectangular area $\Omega =(0;b_1)\times (0;b_2),$ with a border $\partial \Omega =\overline{s},\hspace{0.17em}\hspace{0.17em}s={\Gamma }_1\cup {\Gamma }_2,\hspace{0.17em}\hspace{0.17em}{\Gamma }_1=\left\{b_1\right\}\times (0;b_2)\cup (0;b_1)\times \left\{b_2\right\},\hspace{0.17em}\hspace{0.17em}{\Gamma }_2=\left\{0\right\}\times (0;b_2)\cup (0;b_1)\times \left\{0\right\},$ $n$ is outward normal to $\partial \Omega$, wherein $a=a_1\ge 0$ in the area of ${\Omega }_1$, $a=a_2\ge 0$ in $\Omega \backslash {\Omega }_1$, areas ${\Omega }_1$, ${\Omega }_2$: $\overline{\Omega }={\overline{\Omega }}_1\cup {\overline{\Omega }}_2$, ${\Omega }_1\cap {\Omega }_2=\varnothing$, $\partial {\Omega }_1\cap \partial {\Omega }_2\ne \varnothing$, $P$ is pressure, $K$ is the coefficient of the rigidity of the elastic foundation ($K=0$ in the absence of an elastic foundation), $D=E{h}^3/(12(1-{\sigma }^2))$ is the plate cylindrical stiffness, $h$ is the plate thickness, $E$ is Young’s modulus (tensile modulus), $\sigma \in (0;\hspace{0.17em}1)$ is Poisson’s ratio, $u$ is the desired displacement, $a=K/D$.

• − The mathematical model of the displacement of the points of a plate on an elastic foundation under the action of pressures, where uniform conditions of rigid embedding, hinged support, symmetry, and free support are set on the corresponding edges

$${\Delta }^{2}u+au=f \mathrm{in} \Omega$$

(3)

$$u|{}_{{\Gamma }_0}=\frac{\partial u}{\partial n}|{}_{{\Gamma }_0}=0, u|{}_{{\Gamma }_1}=l_{1}u|{}_{{\Gamma }_1}=0$$

$$\frac{\partial u}{\partial n}|{}_{{\Gamma }_2}=l_{2}u|{}_{{\Gamma }_2}=0, l_{1}u|{}_{{\Gamma }_3}=l_{2}u|{}_{{\Gamma }_3}=0$$

where

$$l_{1}u=\Delta u+(1-\sigma )n_1{n}_2{u}_{xy}-n_2^2{u}_{xx}-n_1^2{u}_{yy}$$

$$l_{2}u=\frac{\partial \Delta u}{\partial \overrightarrow{n}}+(1-\sigma )\frac{\partial }{\partial s}(n_1{n}_2(u_{yy}-u_{xx})+(n_1^2-n_2^2)u_{xy})$$

$$n_1=-\cos (n,x),n_2=-\cos (n,y)$$

Ω is a bounded flat area with piecewise smooth class boundary $C^2$ without self-intersections and self-contacts

$$\partial \Omega =\overline{s},\hspace{0.17em}\hspace{0.17em}s={\Gamma }_0\cup {\Gamma }_1\cup {\Gamma }_2\cup {\Gamma }_3, {\Gamma }_i\cap {\Gamma }_j=\varnothing \mathrm{if} i,j=0,1,2,3$$

${\Gamma }_i,\hspace{0.17em}\hspace{0.17em}i=0,1,2,3-$ is the union of a finite number of disjoint, open subsets of the boundary $\partial \Omega$ from arcs of smooth curves of the class $C^2$, $n$ is outward normal to $\partial \Omega$, $P$ is pressure, $K$ is the coefficient of the rigidity of the elastic foundation ($K=0$ in the absence of an elastic foundation), $D=E{h}^3/(12(1-{\sigma }^2))$ is the plate cylindrical stiffness, $h$ is the plate thickness, $E$ is Young’s modulus (tensile modulus), $\sigma \in (0;\hspace{0.17em}1)$ is Poisson’s ratio, $u$ is the desired displacement, $a=K/D$.

In this paper, we review the solutions for the mathematical models of the theory of elasticity based on the methods of iterative factorizations, fictitious components and other methods.

2. Main Results

2.1. The Poisson and the Sophie Germain-Lagrange Equations

The Poisson equation models the displacement of membranes under pressure, with the potentials of electrostatic fields depending on the density of static charges, stationary temperature distributions from heat sources, fluid flow rates, etc. These models have been studied, for example, in the following papers [1,2,3]. The screened Poisson equation models the movement of membranes under the action of pressures in the presence of elastic foundations [2,4,5].

The Poisson equation is an elliptic partial differential equation. This equation looks like:

$$\mathsf{\Delta }\phi =f,$$

where $\mathsf{\Delta }$ is the Laplace operator, or Laplacian, and $f$ is a real or complex function on some manifold.

In a three-dimensional Cartesian coordinate system, the equation takes the form:

$$\left(\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial z^2}\right)\phi \left(x,y,z\right)=f\left(x,y,z\right).$$

In the Cartesian coordinate system, the Laplace operator is written in the form ${\nabla }^2$ and the Poisson equation becomes:

$${\nabla }^2\phi =f.$$

A mathematical model of the displacements of a rectangular membrane in a variational form is considered. This is a variational problem, a generalized mathematical model of the movement of a rectangular membrane on an elastic foundation under mixed boundary conditions

$$u\in W:\hspace{0.17em}\hspace{0.17em}\mathrm{A}(u,v)=g(v)\hspace{0.17em}\hspace{0.17em}\forall v\in W,\stackrel{⌣}{g}\in {\stackrel{⌣}{W^{\prime }}}_{},$$

(4)

where $W$ the Sobolev function space

$$W=W(\Omega )=\left\{v\in W_2^1(\Omega ):\hspace{0.17em}\hspace{0.17em}{v|}_{{\Gamma }_1}=0\right\}$$

on a rectangular area

$$\Omega =(0;b_1)\times (0;b_2) \mathrm{c} {\Gamma }_1=\left\{b_1\right\}\times (0;b_2)\cup (0;b_1)\times \left\{b_2\right\}$$

$A\left(u,\nu \right)$ is the bilinear form

$$A(u,v)={\displaystyle \underset{\Omega }{\int }(u_x{v}_x+u_y{v}_y+\kappa uv)d\Omega }$$

and given the constants $b_1,\hspace{0.17em}\hspace{0.17em}{b}_2>0,\hspace{0.17em}\hspace{0.17em}\kappa \ge 0.$

Based on [4,6,7], a solution to each problem in (4) exists and is unique if $g\left(v\right)={\displaystyle \underset{\Omega }{\int }fvd\Omega ,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}$ ${\Gamma }_2=\left\{0\right\}\times (0;b_2)\cup (0;b_1)\times \left\{0\right\},$ where $f$ is given, $u$ is the desired function.

In the theory of elasticity, the Sophie Germain-Lagrange equation is a biharmonic equation that is used in the mathematical modeling of pressure plate movements. Similar equations form the basis of mathematical models of displacements of plates under pressure on elastic foundations. Such models were studied, for example, in [2,5,8]. A homogeneous biharmonic equation with inhomogeneous boundary conditions is used in the mathematical modeling of plane problems in the theory of elasticity.

It is customary to call thin plates structural elements in which one dimension (thickness) is significantly less than the other two characteristic dimensions (the length of the sides of the plate). The transverse load $q\left(x,y\right)$ causes the plate to bend, which is accompanied by the appearance of transverse displacements or deflections $w$. In most cases, the problem of calculating plates during bending is reduced to determining deflections when the corresponding boundary (boundary) conditions are met.

In the theory of elasticity, the Sophie Germain–Lagrange equation is used in the mathematical modeling of plate displacement under pressure. This equation in the Cartesian coordinate system has the following form:

$$D{\nabla }^2{\nabla }^{2}w\left(x,y\right)=q\left(x,y\right),$$

or

$$D{\left(\frac{{\partial }^{4}w}{\partial x^4}+\frac{{\partial }^{4}w}{\partial x^2\partial y^2}+\frac{{\partial }^{4}w}{\partial y^4}\right)}^{}=q\left(x,y\right),$$

where $D$ is the bending stiffness of the plate.

2.2. Mathematical Models of Elasticity Theory

When considering various options for the given boundary conditions at the edges of the plates, a lot of mathematical models arise. These models often differ significantly in the methods of possible solutions. The complexity of solving problems arising in these models can also be different. Addressing such issues can be reduced to solving problems, for example, arising in mathematical models of hydrodynamics and nonlinear models of the theory of elasticity [2,9].

Let us note the mathematical models of the displacements of rectangular membranes, with and without an elastic base fixed on two adjacent sides under the action of pressures. The models are built on the basis of the Poisson equations and the screened Poisson equation in a rectangular domain, where the main homogeneous boundary condition is set on two adjacent sides, and the natural homogeneous boundary condition is satisfied on the other two sides. Note that the solution to each problem from the indicated boundary value problem is unique [4,6].

We also note the mathematical model of the displacements of a rectangular plate on an elastic foundation under the action of pressures, where on two adjacent sides there are homogeneous conditions for hinged support, and on the other two sides there are homogeneous symmetry conditions. This is a fourth-order elliptic equation in a rectangular region, where the first main homogeneous boundary condition is set on two adjacent sides and the second main homogeneous boundary condition is set on the other two sides, and the other corresponding boundary conditions are naturally homogeneous. This mathematical model of the displacements of a rectangular plate is considered in a variational form, a generalized mathematical model of the displacement of a rectangular plate on an elastic foundation under mixed boundary conditions

$$u\in V:\hspace{0.17em}\hspace{0.17em}\Lambda (u,v)=g(v)\hspace{0.17em}\hspace{0.17em}\forall v\in V,\stackrel{⌣}{g}\in \stackrel{⌣}{V^{\prime }},$$

(5)

where $V$ is the Sobolev function space

$$V=V(\Omega )=\left\{v\in W_2^2(\Omega ):\hspace{0.17em}\hspace{0.17em}{v|}_{{\Gamma }_1}=0,\frac{\partial v}{\partial n}|{}_{{\Gamma }_2}=0\right\}$$

in the region $\Omega =(0;b_1)\times (0;b_2)$ with border $\partial \Omega =\overline{s},\hspace{0.17em}\hspace{0.17em}s=,{\Gamma }_1\cup {\Gamma }_2,\hspace{0.17em}\hspace{0.17em}{\Gamma }_1=\left\{b_1\right\}\times (0;b_2)\cup (0;b_1)\times \left\{b_2\right\},\hspace{0.17em}\hspace{0.17em}{\Gamma }_2=\left\{0\right\}\times (0;b_2)\cup (0;b_1)\times \left\{0\right\},\hspace{0.17em}$ $n$ is outward normal to $\partial \Omega$, bilinear form

$$\Lambda (u,v)={\displaystyle \underset{\Omega }{\int }(\sigma \Delta u\Delta v+(1-\sigma )(u_{xx}{v}_{xx}+2u_{xy}{v}_{xy}+u_{yy}{v}_{yy})+auv)d\Omega ,}$$

where in $a=a_1$ in the region ${\Omega }_1$, $a=a_2$ in $\Omega \backslash {\Omega }_1$, areas ${\Omega }_1$, ${\Omega }_2$: $\overline{\Omega }={\overline{\Omega }}_1\cup {\overline{\Omega }}_2$, ${\Omega }_1\cap {\Omega }_2=\varnothing$, $\partial {\Omega }_1\cap \partial {\Omega }_2\ne \varnothing$, given constants, $\sigma \in (0;\hspace{0.17em}1)$, $b_{1`},b_2\in \left(0;+\infty \right)$, $a_{1`},a_2\in \left[0;+\infty \right)$.

It can be noted [4,6] that

$$\exists c_1,c_2\in \left(0;+\infty \right):c_1{\Vert v\Vert }_{W_2^2(\Omega )}^2\le \Lambda (v,v)\le c_2{\Vert v\Vert }_{W_2^2(\Omega )}^2\forall \stackrel{⌣}{v}\in \stackrel{⌣}{V}$$

and, consequently, a solution to problem (5) exists and is unique [4,6].

If f is a given and u is a desired, sufficiently smooth functions, and

$$g(v)=(f,v), \mathrm{where} (f,v)={\displaystyle \int fvd\Omega }$$

then problem (5) yields a fourth-order elliptic equation under mixed and homogeneous boundary conditions

$${\Delta }^{2}u+au=f,u{|}_{{\Gamma }_1}=\Delta u{|}_{{\Gamma }_1}=0,\frac{\partial u}{\partial \overrightarrow{n}}{|}_{{\Gamma }_2}=0.$$

The question of the existence and uniqueness of a solution to such a problem was considered by Zh.P. Aubin [6], L.A. Oganesyan, L.A. Rukhovets [4] and others.

Of interest are mathematical models of the displacements of plates on elastic foundations under the action of pressures, where uniform conditions of rigid embedding, hinged support, symmetry, and free support are set on the corresponding edges. These models are described by fourth-order elliptic equations in bounded domains, on a plane under the corresponding boundary conditions of these four types and their various combinations [4,6,10]. Boundary value problems are considered under the usual assumptions that ensure the existence and uniqueness of a solution for each problem.

A mathematical model of plate displacements in a variational form is considered. This is a generalized mathematical model of plate displacements on an elastic foundation under homogeneous mixed boundary conditions, i.e., homogeneous conditions of rigid attachment, hinged support, symmetry, and free support

$$u\in H: \mathsf{\Lambda }\left(u,\nu \right)=g\left(\nu \right) \forall \nu \in H, g\in H^{\prime },$$

(6)

where u is the desired function; $H$ is the Sobolev function space

$$H=H(\Omega )=\left\{v\in W_2^2(\Omega ):\hspace{0.17em}\hspace{0.17em}{v|}_{{\Gamma }_0\cup {\Gamma }_1}=0,\hspace{0.17em}\hspace{0.17em}\frac{\partial v}{\partial n}|{}_{{\Gamma }_0\cup {\Gamma }_2}=0\right\}$$

on a limited flat area $\Omega$ with piecewise-smooth class boundaries without self-intersections and self-tangency

$$\partial \Omega =\overline{s},\hspace{0.17em}\hspace{0.17em}s={\Gamma }_0\cup {\Gamma }_1\cup {\Gamma }_2\cup {\Gamma }_3$$

$${\Gamma }_i\cap {\Gamma }_j=\varnothing \mathrm{if} i\ne j, i,j=0,1,2,3$$

${\Gamma }_i,\hspace{0.17em}\hspace{0.17em}i=0,1,2,3-$ is the union of a finite number of disjoint, open subsets of boundaries $\partial \Omega$ from arcs of smooth curves of the class $C^2$, $n$ is outward normal to $\partial \Omega$, and bilinear forms with constants $a\in \left[0;\hspace{0.17em}+\infty \right)$, $\sigma \in (0;\hspace{0.17em}1)$:

$$\Lambda (u,v)={\displaystyle \underset{\Omega }{\int }(\sigma \Delta u\Delta v+(1-\sigma )(u_{xx}{v}_{xx}+2u_{xy}{v}_{xy}+u_{yy}{v}_{yy})+auv)d\Omega }$$

For problem (6), the assumptions ensuring the existence and uniqueness of its solution to the problem are quite common, e.g., [4,6],

$$\exists c_1,c_2\in \left(0;+\infty \right):c_1{\Vert v\Vert }_{W_2^2(\Omega )}^2\le \Lambda (v,v)\le c_2{\Vert v\Vert }_{W_2^2(\Omega )}^2\hspace{1em}\forall v\in H$$

In the research, reductions of mathematical models in a variational form to mathematical models in a discrete form are used, which preserves the properties of the original models at the difference level based on the finite elements [11,12,13,14,15,16,17].

The construction of optimal, almost optimal, or logarithmically optimal methods, in terms of the number of arithmetic operations for finding solutions with predetermined sufficiently small errors, becomes much more complicated with an increase in the orders of the equations [18]. The numerical solution of the biharmonic equation and similar ones, even in the square region under the main boundary conditions, has not been studied in sufficient detail [19]. Approaches based on the reduction of the solutions to boundary value problems to the minimization of the corresponding functionals obtain practically effective numerical methods [20]. Despite the linear nature of the biharmonic equation, the integration of these problems causes a number of difficulties [21]. However, the numerical solutions of even more complex problems can be reduced to the numerical solution of problems such as those arising in nonlinear models of the theory of elasticity.

Given that these fourth-order problems are recognized as problematic and new approaches are required to solve them, building efficient and programmable methods for calculating the displacements of membranes and plates on elastic foundations under various homogeneous boundary conditions, is required. Efficient and programmable methods are understood as methods that are asymptotically optimal, almost optimal, or logarithmically optimal in terms of the number of arithmetic operations needed to solve them [18].

In this work, the mathematical modeling of membrane and plate displacements is simply reduced to the numerical solutions of Poisson and Germain–Lagrange type equations and, ultimately, to solutions of simple systems of linear algebraic equations with triangular matrices that have no more than three non-zero elements in each line [22,23].

For the reduction of complex problems in mathematical physics, these are usually reduced to algebraic equations of a certain structure [14].

While maintaining a logically simple research methodology, when problematic tasks are considered that require new knowledge and approaches for their solution, research is placed in the area of formally more general tasks. These problems are then effectively solved based on the proposed iterative methods.

If statistical methods are better suited for very complex problems, numerical iterative methods for complex problems, and direct methods for simple problems, then for the problems under consideration it is preferable to build numerical iterative solution methods [24]. For the approximate solution of the difference and variational-difference analogues of the problems, iterative methods are often used, which occupy an intermediate position between direct and statistical methods.

There are methods for solving similar equations even on the basis of a statistical approach, for example, using the Monte Carlo method [25,26,27]. For elliptic equations, the Monte Carlo method is not optimal or logarithmically optimal in terms of the number of arithmetic operations [14,18]. However, obtaining analytical and practically acceptable direct solutions of such equations in the final form is impossible.

When solving these equations numerically, if using a system of linear algebraic equations with $N$ unknowns variables, then in the direct Gauss method, an order of arithmetic operations $N^3$ is required to solve it. Such computational costs will not be economical, despite the rapid progress in computer technology [24]. The optimal number of arithmetic operations for solving such a problem is of the order of N. The asymptotic solution is optimal in terms of the number of arithmetic operations, since it is unimprovable [18].

2.3. Approximation of Step Functions in Problems of Mathematical Modeling

In the theory of elasticity, methods of approximating piecewise linear and generalized functions have been proposed for modeling jump processes [28,29,30,31,32,33]. These methods do not have the disadvantages inherent in expansions in a Fourier series. They are based on the use of trigonometric expressions, but in the form of recursive functions.

Let us take the following step function as an example, which will allow us to explain the main idea of the proposed approximation methods:

$$f_0\left(x\right)=\mathrm{sign}\left(\sin x\right)$$

(7)

We chose this function because it explains well the difference between the usual expansion of a function in a Fourier series and the approximation of a step function using the proposed method. The selected function is often used to illustrate the procedure for expanding a function using trigonometric series. Note that the expansion of functions into Fourier series has not only positive, but also negative features. We note, for example, the presence of the Gibbs effect, according to which, for a certain class of functions, the approximation error using Fourier series does not tend to zero with an increase in the number of terms in the expansion. The proposed approximation methods do not have the disadvantages of Fourier series, and are based on the expansion of the original function using recursive sequences of periodic trigonometric functions. For example, the approximation dependence of the selected step function (2.4) according to the proposed method has the form:

$$\left\{f_n(x)\hspace{0.17em}|{ f}_n(x)=\sin \left((\pi /2)\cdot f_{n-1}(x)\right),\hspace{0.17em}{f}_1(x)=\sin x;\hspace{0.17em}n-1\in \mathit{N}\right\}\subset C^{\infty }[-\pi ,\pi ]$$

(8)

The proposed procedure ensures fast convergence of the iterative sequence to the original function. In this case, the approximation error is small, even when using a relatively small number of iterations. It is important to note that the negative Gibbs effect is completely absent in the proposed approximation method.

The proposed approximating iterative method is characterized by some features that require additional explanation. Consider the approximating sequence (8) on the interval $\left[0,\hspace{0.17em}\pi /2\right]$, taking into account that periodic functions $f_n(x)$ and $f_0(x)$ with period $2\pi$ are odd, and periodic functions $f_n(x+\pi /2)$ and $f_0(x+\pi /2)$ are even.

Consider a sequence of functions $\left\{f_n(x)\right\}\subset L_2[0,\pi /2]$ and the function $f_0(x)\in L_2[0,\pi /2]$. Because the functions $f_n(x)$ are bounded and monotonic on the segment $[0,\pi /2]$, we get $\underset{\mathit{n}\in \mathit{N}}{\sup } \underset{\mathit{x}\in [0,\hspace{0.17em}\pi /2]}{\sup }\left|f_{\mathit{n}}(x)\right|=1<\infty$ and $\underset{\mathit{n}\in \mathit{N}}{\sup }\underset{0}{\overset{\pi /2}{\mathit{Var}}}{f}_n=1<\infty$, therefore we can apply Helly’s theorem, according to which it is possible to select a subsequence from the sequence $\left\{f_n(x)\right\}$ that converges to some function $f$ at each point $[0,\hspace{0.17em}\pi /2]$, so that, $\underset{0}{\overset{\pi /2}{Var}}f\le \underset{n\to \infty }{\overset{\_\_\_}{\lim }}\underset{0}{\overset{\pi /2}{Var}}{f}_n$. It can be shown that the function $f$ can be as such a function $f_0(x)$.

Statement 1. 

There is a pointwise convergence of the sequence of functions$f_n(x)$to the original function $f_0(x)$, and this convergence is not uniform.

We can prove this statement.

In our case it is obvious that $f_n(x)-f_0(x)=0,\hspace{0.17em}\forall n\in \mathit{N}$ at the points $x=0$ and $x=\mathit{\pi }/2$. That is why, for these points we have $f_n(x)\underset{n\to \infty }{\to }{f}_0(x)$ because $\forall \epsilon >0\hspace{0.17em}\exists n^{\ast }\in \mathit{N}\hspace{0.17em}\forall n:n>n^{\ast }\Rightarrow \hspace{0.17em}\left|f_n(x)-f_0(x)\right|=0<\epsilon$. For instance, it is possible to have $n^{\ast }=1$.

We have $\sin x>(2/\pi )\cdot x,\hspace{0.17em}\forall x\in (0,\hspace{0.17em}\pi /2)$, therefore we get $f_n(x)=\sin \left((\pi /2)\cdot f_{n-1}(x)\right)>f_{n-1}(x)>\dots >f_1(x)>0$ that is right for any points from the interval $x\in (0, \pi /2)$. The sequence of functions $f_n(x),\hspace{0.17em}\forall x\in (0,\hspace{0.17em}\pi /2)$ is positive, increasing, and bounded. Therefore, for this sequence there is a finite limit. Let us denote the limit by $\underset{n\to \infty }{\lim }{f}_n(x)=A\in \mathit{R}$. So, we have

$$A=\underset{n\to \infty }{\lim }\sin ((\pi /2)\cdot f_{n-1}(x))=\sin ((\pi /2)\cdot \underset{n\to \infty }{\lim }{f}_{n-1}(x))=\sin ((\pi /2)\cdot A)$$

Based on this expression, we can claim that $A=0$ or $A=1$. We know that our sequence is positive and increasing. Therefore $A=1=f_0(x)$, ∀x∈(0, π/ 2). Further, for this segment we have that $f_n(x)\underset{n\to \infty }{\to }{f}_0(x)$. Earlier, we noted the convergence of our sequence of functions at the points $x=0$ and $x=\pi /2$. Therefore, we conclude that $f_n(x)\underset{n\to \infty }{\to }{f}_0(x),\hspace{0.17em}\forall x\in [0,\hspace{0.17em}\pi /2]$. Moreover, the proven convergence is only pointwise, but it is not uniform, since the original function $f_0(x)$ is not continuous on the considered segment $\left[0,\hspace{0.17em}\pi /2\right]$. Q.E.D.

Statement 2. 

There is a convergence of a sequence of approximating functions$f_n(x)$in the norm to a given function $f_0(x)$, if we consider the space of measurable functions $L_1[0,\pi /2]$and the Hilbert space $L_2[0,\pi /2]$.

We can prove this statement.

Consider the sequence of minorant functions with respect to the sequence of given functions $f_n(x)$

$$\left\{{\eta }_n(x)\hspace{0.17em}|\hspace{0.17em}{\eta }_n(x)=(2/\pi )\cdot \mathrm{arctg}(n\pi );\hspace{0.17em}n\in \mathit{N}\right\}\subset C^{\infty }[0,\pi /2]$$

We can show that $f_n(x)\ge {\eta }_n(x), \forall n\in \mathit{N},\hspace{0.17em}\forall x\in [0,\hspace{0.17em}\mathit{\pi }/2]$. It is not difficult to see that the set of discontinuity points of the original function $f_0(x)$ has a measure equal to zero. Given that the functions $f_n(x)$ and ${\eta }_{\mathit{n}}(x)$ are bounded and non-negative in the segment, then we consider we have in space $L_1[0,\hspace{0.17em}\pi /2]$:

$$\left|\left|f_0(x)-f_n(x)\right|\right|={\displaystyle \underset{0}{\overset{\pi /2}{\int }}(1-f_n(x))\mathit{dx}}\le {\displaystyle \underset{0}{\overset{\pi /2}{\int }}(1-{\eta }_n(x))\mathit{dx}}=\frac{\pi }{2}-\mathrm{arctg}\frac{\pi n}{2}+\frac{1}{\pi n}\cdot \ln \left(1+{(\pi n)}^2/4\right)$$

Because $\underset{n\to \infty }{\lim }\left(\frac{\pi }{2}-\mathrm{arctg}\frac{\pi n}{2}+\frac{1}{\pi n}\cdot \ln \left(1+{(\pi n)}^2/4\right)\right)=0$, we have

$$\left|\left|f_0(x)-f_n(x)\right|\right|\underset{n\to \infty }{\to }0$$

In the same way, we can prove the convergence in the norm of a sequence of functions $f_n(x)$ to the initial function $f_0(x)$ in the Hilbert space $L_2[0,\pi /2]$. Q.E.D.

We have shown that the sequence of approximating functions $f_n(x)$ is fundamental in the spaces $L_1[-\pi ,\pi ]$ and $L_2[-\pi ,\pi ]$. Note that this sequence $f_n(x)$ is not fundamental in the space $C[-\pi ,\pi ]$.

Let us call the function $f_1(x)$ initial or angular. With this function we do not use the sine, but some other, and not necessarily periodic, function. For example, using the iterative procedure (2) and fulfilling the conditions $\left|f_1(x)\right|<2$, we have $\underset{n\to \infty }{\lim }{f}_n(x)=sign(f_1(x))$. It follows from this conclusion that any step function can be approximated using the proposed methods. For example, in the case of a step function

$$f(x)=\{\begin{array}{l}h,\hspace{0.17em}x\in (x_1,x_2),\\ 0,\hspace{0.17em}x\notin (x_1,x_2)\end{array}$$

(9)

the following can be chosen as the initial function $f_1(x)=\exp (1-{(ax+b)}^2)-1$. Using the following condition $f_1(x_1)=f_1(x_2)=0$, we can find that $a=2/(x_1-x_2);\hspace{0.17em}\hspace{0.17em}b=(x_1+x_2)/(x_2-x_1)$. Taking the founded values of the coefficients $a$ and $b$, we can write the sequence

$$\left\{f_n(x)\hspace{0.17em}|\hspace{0.17em}{f}_n(x)=(h/2)\cdot (1+\sin {\mathsf{\phi }}_{\mathit{n}}(x)),\hspace{0.17em}{\mathsf{\phi }}_{\mathit{n}}(x)=(\pi /2)\cdot \sin {\mathsf{\phi }}_{\mathit{n}-1},\hspace{0.17em}{\mathsf{\phi }}_1(x)=(\pi /2)\cdot f_1(x),\hspace{0.17em}n-1\in \mathit{N}\hspace{0.17em}\right\}$$

which converges to the step function $f(x)$. Therefore, any step function with values $h_i$ within the segment $(x_{1i},x_{2i})$ we can approximate using the sum of the sequences ${\displaystyle \sum _{i=1}^k{\left\{f_n(x)\right\}}_i}$.

Statement 2 has a universal character and can be applied to the analytical approximation of any step function, since we can write an arbitrary periodic step function using a linear combination $f\left(x\right)={\displaystyle \sum }_{i=1}^k{h}_i·f_{0i}\left(x\right), h_i\in R,$ and, if necessary, shifting in phase and along the y-axis the functions $f_{0i}\left(x\right)=\mathrm{sign}\left(\sin \left(l_{i}x-x_i\right)\right),l_i,x_i\in R.$Based on the statement 2, we can assert that in the spaces $L_1\left[-\pi ,\pi \right]$and $L_2\left[-\pi ,\pi \right]$ there is convergence $\Vert f_{0i}\left(x\right)-f_{ni}\left(x\right)\Vert \underset{n\to \infty }{\to }0$, $\forall i$, so the function $f_n\left(x\right)={\displaystyle \sum }_{i=1}^k{h}_i·f_{ni}\left(x\right)$converges to the function $f\left(x\right)$ in norm, because

$$\Vert f\left(x\right)-f_n\left(x\right)\Vert =\Vert {{\displaystyle \sum }}_{i=1}^k{h}_i·f_{0i}\left(x\right)-{{\displaystyle \sum }}_{i=1}^k{h}_i·f_{ni}\left(x\right)\Vert \le {{\displaystyle \sum }}_{i=1}^k\left|h_i\right|·\Vert f_{0i}\left(x\right)-f_{ni}\left(x\right)\Vert \underset{n\to \infty }{\to }0.$$

A review of these and other methods is given in [34].

2.4. Multigrid Methods for Solving the Biharmonic Equation

Increasing requirements for the accuracy of the results of mathematical modeling leads to an increase in the number of linear algebraic equations, the complication of their form and an increase in the number of arithmetic operations to solve them [24]. Consequently, there is a practical need for efficient, economical numerical methods that are optimal, or at least almost optimal in terms of computational costs.

Without presuming complete coverage of all methods for solving the biharmonic equation, some of the approaches currently being studied can be noted. There is an extensive literature where mathematical models are considered, which are usually based on partial differential equations of no higher than the second order, while the biharmonic equation is less common. There is also a desire for its effective solution [35]. Multigrid methods are often used [36]. Multigrid iterative algorithms are used for the discrete analogue [37].

There is a need for methods to reduce the order of the biharmonic equation [38]. Sorokin [39] reduces the solution of a biharmonic equation in a rectangle, when the conditions of hinge support are given on three sides, and the conditions of free support on the remaining side reduces to solving a sequence of problems for the same equation, but with boundary conditions of hinge support, which allow splitting the problem into two Dirichlet problems for the Poisson equation. When the problem is continued across the boundary, with natural boundary conditions in methods of the type of fictitious components, computational costs are usually optimal if the problems that arise at each step of the iterative process are optimally solved, as in this paper. Sorokin [40,41] also reduces the numerical solution of the biharmonic equation to the numerical solution of the Dirichlet problems for the Poisson equation. He proposed an iterative process for the numerical solution of the biharmonic equation in a rectangular domain, with principal boundary conditions on the boundary which is not logarithmically optimal [40].

There is a well-known approach that does not claim to be logarithmic optimality [42,43]. In these previous and subsequent works, a fourth-order elliptic boundary value problem in a domain is reduced to second-order elliptic problems in the same domain, solving some integral equation on the boundary of the domain. Karachik, Antropova [44,45,46] constructed polynomial solutions for the Dirichlet problem for a biharmonic equation in a ball. Potapov [47] uses the reduction of the solution of a biharmonic equation to the solution of a system of ordinary differential equations.

The number of works on this topic is increasing, and this indirectly indicates that the numerical solution of such problems has certain difficulties, and problems remain, i.e., their solutions require new approaches. The presence of a sufficiently large number of works on the solution of the Germain–Lagrange plate equation within the framework of possibly irrational logic indicates the importance of these areas of research.

The much larger number of works on the solution of the Poisson equation, which is often reduced to the solution of the biharmonic equation, may indicate both the advantages and possible shortcomings in the methods for solving it, and, consequently, the need for, and possibility of, their improvement.

For numerical solutions of elliptic boundary value problems of the second order in rectangular areas, there are direct methods that are optimal or logarithmically optimal in terms of the number of arithmetic operations [18,48,49,50,51,52,53,54,55]. Direct marching methods are optimal, but they can be numerically unstable to rounding errors in computer calculations [44,48,49,52]. Unfortunately, the issues of stability with respect to rounding errors for these methods are also quite acute [18].

In general cases, the equations of Poisson and Germain–Lagrange plates can hardly be optimally finally solved, given that these are not two problems, but a set of problems in different areas and under different boundary conditions. Therefore, questions arise about the possibility of a successful combination of efficiency and optimality in the number of arithmetic operations of the solution methods and the ease of their computation; the computational stability of the methods to rounding errors is assumed. There are already approaches for the error-free solution of systems of linear algebraic equations [56].

In the one-dimensional case, there are second-order elliptic equations, operators that factorize under certain boundary conditions, and their finite-difference analogs [14]. In the two-dimensional case, for second-order elliptic equations, the Buleev method of incomplete factorization was developed in [14,18,57,58,59,60,61]. This method has a comparable number of operations with the logarithmically optimal variants of the alternating directions method [14].

Novikov and Dynnikov discovered the complex factorization of the Laplace operator, together with its difference analogue on triangular grids, and pointed out the absence of factorization on commonly used rectangular grids [62,63,64]. However, the possibility of complex factorization when approximating the Laplace operator only on triangular grids was not known to the authors, which allowed them to obtain an iterative complex factorization when approximating the Laplace operator on rectangular grids [65,66] for the problems indicated in [67,68], and use it for their asymptotically optimal solution.

Factorized models in problems of mathematical physics were considered in [69,70]. Matsokin [71] implements the idea of factorization in the two-dimensional case of a difference operator that arises when approximating a fourth-order elliptic-type equation in a rectangle, on the boundary of which the hinged support conditions are specified. If, at each step of the iterative process in this paper, we apply not the logarithmically optimal direct method, as expected, but the method of iterative factorizations proposed in [65] for a problem with mixed boundary conditions, then we get a gain in the number of arithmetic operations.

When obtaining effective methods, the spectral, energy equivalence of operators obtained by approximating elliptic equations under certain boundary conditions by the finite element method and the finite difference method is often used [18,72,73,74,75,76]. Dyakonov [18] constructs a logarithmically optimal method for a model elliptic problem of the fourth order in a rectangle under the main boundary conditions. In this paper [18], basic model elliptic problems in a rectangle are considered under mixed boundary conditions, and these problems are asymptotically optimally solved in [18] based on iterative factorization methods [65,66,67,68].

For sufficiently arbitrary domains, methods for solving fourth-order elliptic equations are constructed. These methods are used to solve equations with basic boundary conditions. They are logarithmically optimal when continuing across the boundary, and asymptotically optimal when continuing the solution of the problem across the boundary with natural boundary conditions. The solutions obtained are modifications of the corresponding known methods using fictitious components. The finite element method is used for parabolic completions [6]. Margenov, Lazarov [77] performed work using parabolic splines for the numerical solution of fourth-order elliptic equations, where the computational cost is clearly not optimal.

A popular approach to solving elliptic equations is based on the Schwartz method [24,78,79,80,81,82,83]. The original problem is effectively reduced to a series of problems which are optimally solved for second-order equations by marching methods.

There are a large number of works on iterative solutions of fourth-order elliptic equations by Korneev, for example, [84]. A logarithmically optimal method is constructed for a fourth-order equation under natural boundary conditions, based on the methods of fictitious components and the fast discrete Fourier transform [85]. The methods of fictitious components are iterative processes for solving systems of difference and variational difference equations. The fast discrete Fourier transform [85] is an algorithm to accelerate the calculation of the discrete Fourier transform, which allows you to get the result in less than O(N²) time (required for direct, formula-by-formula calculation). For similar problems, the iterative factorization methods proposed in [65,66,67,68], in combination with the modified method of fictitious components, give asymptotically optimal methods.

The method of fictitious components was proposed, investigated and optimized, in theoretical terms, in [80,81,82,86,87,88,89,90,91,92]. The use of modifications of this technique [65,66,67,68] and the methods of iterative factorizations makes it possible to obtain numerical iterative methods that are efficient in terms of the number of arithmetic operations for calculating the displacements of plates.

The method of fictitious components was developed in [91], which used operators that give a norm-preserving extension of discrete functions. The method is asymptotically optimal in terms of the number of arithmetic operations if we use the iterative factorization methods proposed in [65], not the fast discrete Fourier transform. Thus, the methods of iterative factorizations can be more effectively used in combination with the fictitious space method for the numerical solution of elliptic boundary value problems of the second order, in arbitrary areas of complex geometric shape.

In [65,66,67], the main model problems of the second and fourth orders in a rectangular domain are considered, and new efficient methods for their solution are proposed. The proposed solutions are based on the special properties of these problems. In [65], unstable replacements for optimal marching methods are proposed. The works [52,55,56,59] describe efficient methods of iterative factorization. At the final stages in [93], the author proposes modifications of the methods proposed and developed in [78,79,80,81,82] for reducing solutions of fourth-order problems in arbitrary domains, to solving the indicated basic model problems in a rectangular domain. In a number of cases [93,94,95,96,97,98,99,100], for fourth-order equations under basic boundary conditions, in contrast to the second-order equations considered in [91], only logarithmic optimality takes place. This preserves the simple implementation of the methods. The difference is no longer inherent in the modifications of the methods used, but the last difference between optimality and almost optimality can be sacrificed for the sake of the simple practical implementation and automation of the application of numerical methods. In accordance with [14], under the natural assumption that the modulus of the logarithm of the relative error of the iterative process of the same order as the logarithm of the number of unknowns of the system of linear algebraic equations being solved, the iterative processes of fictitious component methods are transformed from asymptotically optimal methods into logarithmically optimal methods.

2.5. Modified Mathematical Models in the Theory of Elasticity

New approaches are completed by obtaining efficient numerical methods based on replacing the original mathematical models with their fictitious extensions. The theoretical results have been experimentally confirmed by calculations. One can note the revival of the direction of methods of fictitious regions, unknowns, and components and spaces in [100,101].

In [102], the stressed state of an eccentric ring made of an elastic material is studied. Loading is performed by concentrated forces symmetrically along the outer boundary. The Airy stress function written in bipolar coordinates was used. The elastic potential corresponding to a given load was considered for a compact disk in bipolar coordinates. An analytical expression of the elastic potential in a Fourier series was carried out. Boundary conditions were imposed by adding an auxiliary potential. The number of terms in the Fourier expansion is subject to significant change, depending on the final form of the total potential. The results were confirmed using finite element software, and using the photo-elastic method in experimental studies. The article investigated six loading options for two rings with different geometric dimensions, and all the analyzed cases are characterized by good convergence of analytical, numerical, and experimental results. Using analytical calculations, the effect of stress concentration on the inner hole was found. As the eccentricity of the inner hole decreases, the integrals describing the total elastic potential have decreasing convergence near the inner boundary.

To obtain a general solution to the plane problem of elasticity theory, the Airy stress function $U\left(x.y\right)$ was found in the form of a biharmonic function, the property of which is expressed by the relation:

$$∆∆U\left(x,y\right)=0.$$

A particular solution of the problem was obtained by imposing boundary conditions. In this case, various possible cases were considered: taking into account only stresses, only displacements, or superimposed stresses on the one hand and superimposed displacements on the other. To obtain the values of stresses and displacements, the derivatives of the stress function were used. Stresses in Cartesian coordinates have the form:

$${\sigma }_x=\frac{{\partial }^{2}U}{\partial y^2}; {\sigma }_y=\frac{{\partial }^{2}U}{\partial x^2}; {\tau }_{xy}=-\frac{{\partial }^{2}U}{\partial x\partial y} ,$$

and the displacements u and ν correspond to the dependencies:

$$\lambda \theta +2\mu \frac{\partial u}{\partial x}=\frac{{\partial }^{2}U}{\partial y^2}; \lambda \theta +2\mu \frac{\partial v}{\partial y}=\frac{{\partial }^{2}U}{\partial x^2} ,$$

where

$$\theta =\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}$$

is the bulk deformation and the coefficients λ and µ are the Lamé constants.

Under the given boundary conditions, the correct choice of the coordinate system is important for solving a partial differential equation; it should provide the simplest form of boundary curves. Ideally, the boundary curves should be the coordinate curves of the selected reference system.

In the case of a homogeneous isotropic elastic material, the boundaries of the eccentric ring are two non-concentric circles. In this case, the bipolar coordinate system is the most adequate. The description of the relationship between bipolar coordinates (α, β) and Cartesian coordinates (x, y) is given by any of the following systems:

$$x=\frac{a·\sin \beta }{\cos h\alpha -\cos \beta };y=\frac{a·\sin h\alpha }{\cos h\alpha -\cos \beta };$$

$$\alpha +i\beta =\ln \frac{x+i\left(y+a\right)}{x+i\left(y-a\right)}=\ln \frac{z+ia}{z-ia};$$

$$x^2+{\left(y-\frac{a}{\tan h\alpha }\right)}^2={\left(a/\sin h\alpha \right)}^2;$$

$$y^2+{\left(x-\frac{a}{\tan \beta }\right)}^2={\left(a/\sin \beta \right)}^2.$$

The monograph [103] describes the main provisions of the theory of ill-posed abstract integro-differential equations, Volterra. This theory of linear integro-differential equations has received significant development, especially in the last 30 years. A significant part of the monograph refers to the study of various types of abstract multi-term fractions.

Similar with the previous monograph, the work [104] is devoted to the study of the theory of abstract Volterra equations. As noted earlier, this theory has been rapidly developing over the past three decades. Basically, such interest in this theory is explained by its wide applications to problems of mathematical physics. One can note the applications of that theory in such areas as viscoelasticity, thermal conductivity in materials with memory, electrodynamics, and others. The theory of abstract Volterra equations provides useful methods and tools for solving problems that arise in these areas. The monograph [104] describes a significant number of interesting phenomena that are not detected using differential equations, but are observed on specific examples of the Volterra equations. This makes it possible to improve research to improve the understanding of real phenomena, a more accurate reflection of reality. At present, there are several monographs devoted to the development of research into the Volterra equations. These monographs do not consider linear problems in infinite dimensions. The noted monograph [104] closes this shortcoming. The purpose of the monograph is to systematically and completely describe the linear theory in its current state. The monograph reflects relevant results in the field of abstract Volterra equations and their wide applications. In addition, this monograph notes a number of problems in linear theory that have not yet found their solution and require their detailed consideration.

3. Conclusions

The review of the methods of the mathematical modeling of the theory of elasticity revealed that effective methods have now been built for calculating the displacements of rectangular membranes and plates under the specified boundary conditions. Calculations of plate displacements with the help of modifications of the fictitious component method are effectively reduced to the solutions of the previously mentioned problems in rectangular domains.

This review presents an analysis of articles and a description of recent research in the field of mathematical modeling of problems and processes of elasticity theory using iterative factorizations and fictitious components. To carry out the noted studies, various methods have been developed, which are noted in the review. One can specify, for example, an approximate analytical method of iterative factorizations for calculating displacements of a rectangular plate, a modified method of fictitious components for calculating continuous displacements of plates, efficient numerical methods of iterative factorizations for calculating displacements of rectangular membranes, and other methods. Computer implementation of the algorithms of these methods has confirmed their effectiveness. Particular attention is paid to problematic issues that arise in solving problems of the theory of elasticity. The presented review includes research on the development of methods for reducing the solution of complex problems to solving simple problems. As an example, we can cite methods for lowering the order of differential equations and reducing solutions in complex domains, to solutions in a simple domain using iterative factorization methods and fictitious component methods.

Numerical methods have been developed for calculating the displacements of rectangular membranes and plates under the indicated mixed boundary conditions, and are effective in terms of the number of computational operations.

When modifying the fictitious component method, numerical methods for calculating the displacements of plates are constructed, which are logarithmically optimal for rigid mounting and asymptotically optimal for free support in terms of the number of arithmetic operations.

Based on the results of the mathematical modeling, computer programs have been compiled for calculating the displacements of membranes and plates.