<p>Boundary-contact problems (BCPs) are studied within the frames of</p><p>classical mathematical theory of elasticity and plasticity</p><p>elaborated by Landau, Kupradze, Timoshenko, Goodier, Fichera and</p><p>many others on the basis of analysis of two- and three-dimensional</p><p>boundary value problems for linear partial differential equations.</p><p>A great attention is traditionally paid both to theoretical</p><p>investigations using variational methods and boundary singular</p><p>integral equations (Muskhelishvili) and construction of solutions</p><p>in the form that admit efficient numerical evaluation (Kupradze).</p><p>A special family of BCPs considered by Shtaerman, Vorovich,</p><p>Alblas, Nowell, and others arises within the frames of the models</p><p>of squeezing thin multilayer elastic sheets. We show that</p><p>mathematical models based on the analysis of BCPs can be also</p><p>applied to modeling of the clich\'{e}-surface printing contacts</p><p>and paper surface compressibility in flexographic printing.</p><p>The main result of this work is formulation and complete</p><p>investigation of BCPs in layered structures, which includes both</p><p>the theoretical (statement of the problems, solvability and</p><p>uniqueness) and applied parts (approximate and numerical</p><p>solutions, codes, simulation).</p><p>We elaborate a mathematical model of squeezing a thin elastic</p><p>sheet placed on a stiff base without friction by weak loads</p><p>through several openings on one of its boundary surfaces. We</p><p>formulate and consider the corresponding BCPs in two- and</p><p>three-dimensional bands, prove the existence and uniqueness of</p><p>solutions, and investigate their smoothness including the behavior</p><p>at infinity and in the vicinity of critical points. The BCP in a</p><p>two-dimensional band is reduced to a Fredholm integral equation</p><p>(IE) with a logarithmic singularity of the kernel. The theory of</p><p>logarithmic IEs developed in the study includes the analysis of</p><p>solvability and development of solution techniques when the set of</p><p>integration consists of several intervals. The IE associated with</p><p>the BCP is solved by three methods based on the use of</p><p>Fourier-Chebyshev series, matrix-algebraic determination of the</p><p>entries in the resulting infinite system matrix, and</p><p>semi-inversion. An asymptotic theory for the BCP is developed and</p><p>the solutions are obtained as asymptotic series in powers of the</p><p>characteristic small parameter.</p><p>We propose and justify a technique for the solution of BCPs and</p><p>boundary value problems with boundary conditions of mixed type</p><p>called the approximate decomposition method (ADM). The main idea</p><p>of ADM is simplifying general BCPs and reducing them to a chain</p><p>of auxiliary problems for 'shifted' Laplacian in long rectangles</p><p>or parallelepipeds and then to a sequence of iterative problems</p><p>such that each of them can be solved (explicitly) by the Fourier</p><p>method. The solution to the initial BCP is then obtained as a</p><p>limit using a contraction operator, which constitutes in</p><p>particular an independent proof of the BCP unique solvability.</p><p>We elaborate a numerical method and algorithms based on the</p><p>approximate decomposition and the computer codes and perform</p><p>comprehensive numerical analysis of the BCPs including the</p><p>simulation for problems of practical interest. A variety of</p><p>computational results are presented and discussed which form the</p><p>basis for further applications for the modeling and simulation of</p><p>printing-plate contact systems and other structures of</p><p>flexographic printing. A comparison with finite-element solution</p><p>is performed.</p>

Boundary-contact problems (BCPs) are studied within the frames of classical mathematical theory of elasticity and plasticity elaborated by Landau, Kupradze, Timoshenko, Goodier, Fichera and many others on the basis of analysis of two- and three-dimensional boundary value problems for linear partial differential equations. A great attention is traditionally paid both to theoretical investigations using variational methods and boundary singular integral equations (Muskhelishvili) and construction of solutions in the form that admit efficient numerical evaluation (Kupradze). A special family of BCPs considered by Shtaerman, Vorovich, Alblas, Nowell, and others arises within the frames of the models of squeezing thin multilayer elastic sheets. We show that mathematical models based on the analysis of BCPs can be also applied to modeling of the clich\'-surface printing contacts and paper surface compressibility in flexographic printing. The main result of this work is formulation and complete investigation of BCPs in layered structures, which includes both the theoretical (statement of the problems, solvability and uniqueness) and applied parts (approximate and numerical solutions, codes, simulation). We elaborate a mathematical model of squeezing a thin elastic sheet placed on a stiff base without friction by weak loads through several openings on one of its boundary surfaces. We formulate and consider the corresponding BCPs in two- and three-dimensional bands, prove the existence and uniqueness of solutions, and investigate their smoothness including the behavior at infinity and in the vicinity of critical points. The BCP in a two-dimensional band is reduced to a Fredholm integral equation (IE) with a logarithmic singularity of the kernel. The theory of logarithmic IEs developed in the study includes the analysis of solvability and development of solution techniques when the set of integration consists of several intervals. The IE associated with the BCP is solved by three methods based on the use of Fourier-Chebyshev series, matrix-algebraic determination of the entries in the resulting infinite system matrix, and semi-inversion. An asymptotic theory for the BCP is developed and the solutions are obtained as asymptotic series in powers of the characteristic small parameter. We propose and justify a technique for the solution of BCPs and boundary value problems with boundary conditions of mixed type called the approximate decomposition method (ADM). The main idea of ADM is simplifying general BCPs and reducing them to a chain of auxiliary problems for 'shifted' Laplacian in long rectangles or parallelepipeds and then to a sequence of iterative problems such that each of them can be solved (explicitly) by the Fourier method. The solution to the initial BCP is then obtained as a limit using a contraction operator, which constitutes in particular an independent proof of the BCP unique solvability. We elaborate a numerical method and algorithms based on the approximate decomposition and the computer codes and perform comprehensive numerical analysis of the BCPs including the simulation for problems of practical interest. A variety of computational results are presented and discussed which form the basis for further applications for the modeling and simulation of printing-plate contact systems and other structures of flexographic printing. A comparison with finite-element solution is performed.