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The construction of adjacent double-line tunnels will have mutual influence, and the section convergence deformation of double-hole tunnels is quite different from that of single-hole tunnels. A new displacement boundary condition was proposed based on the characteristics of the deformation of the adjacent double-line tunnel. By introducing the displacement release coefficient η, the displacement generated by the tunnel excavation and the lining construction was considered respectively. According to the theory of complex variable function and alternating method, the semi-analytical solution of the adjacent double-line shield tunnel after excavation was obtained. and the theoretical solution was verified by two field engineering monitoring data. Finally, the geometrical parameters of the tunnel were analysed to obtain the disturbance pattern of ground deformation under the influence of different factors in the construction of a two-lane tunnel. The results show that the lateral deformation of the tunnel has a large effect on the shape and size of the surface settlement curve, however, it has a small effect on the width of the surface settlement trough. Displacement release coefficients have a significant effect on surface settlement, which decreases as the displacement release coefficient increases and vice versa. The greater the depth of the tunnel, the smaller the surface settlement value and the larger the area of influence, and when it exceeds a certain depth, the change in the surface settlement value is relatively small. As d/R increases, the surface settlement curve shape evolves from V-U-W. [ABSTRACT FROM AUTHOR]

In this manuscript, we review the performance of domain decomposition methods (DDMs), implemented as a black-box module integrated with a finite element solver, for modeling materials with complex microstructures. In particular, we study the accuracy and computational cost associated with using the non-overlapping and overlapping Schwarz methods, together with required adjustments for each method to avoid convergence issues. Compared to conventional applications such as fluid–solid interaction, the DDM simulation of the mechanical behavior of materials with complex heterostructures could be a challenging task due to high stress concentrations along subdomain edges intersecting with multiple material interfaces. For linear elastic problems, this could lead to high local errors along sub-domain boundaries and especially at subdomain vertices, which requires meticulous updating of boundary conditions (nodal forces and displacements) along these edges to alleviate the error. However, for nonlinear (elastoplastic) problems, we show that such microstructural features prohibit the convergence of the non-overlapping Schwarz method. The remedy to such convergence difficulties is to implement the overlapping Schwarz method, with a high overlap percentage between adjacent subdomains to achieve a reasonable computational cost. [ABSTRACT FROM AUTHOR]

One of the most relevant areas of research in biomedical anthropology is the establishment of morphometric parameters of the human body, individual organs and structures of the body, in particular cephalometric indicators and indicators of the jaw-facial apparatus. Such research is important from the point of view of practical medicine, first of all improvement of methods of identification of the person in forensic medicine and for creation of a normative database for diagnostics of dental deformities and improvement of consequences of surgical, orthopedic and orthodontic interventions. One of the most accurate and effective methods for determining cephalometric parameters is the method of lateral teleroentgenography. The object: to establish teleroentgenographic indicators that characterize the position of teeth and the profile of facial soft tissues and determine their sexual characteristics in Ukrainian young men and young women with orthognathic occlusion depending on the profiles and types of faces according to Schwarz A. M. Conclusions: the results prove the need to consider not only age, but also sex, type and profile of the face for an individualized approach in determining the normative teleroentgenographic indicators of human, are important in forensic medicine to improve the efficiency of identification and in practical dentistry to assess the condition of the dental apparatus and ensure effective therapeutic and aesthetic results of orthopedic and orthodontic dentistry. [ABSTRACT FROM AUTHOR]

In this paper, we present a maximum norm analysis of an overlapping Schwartz method on non matching grids for a quasi-variational inequality, where the obstacle and the second member depend on the solution. Our result improves and generalizes some previous results. [ABSTRACT FROM AUTHOR]

Random sampling has been used to find low-rank structure and to build fast direct solvers for multiscale partial differential equations of various types. In this work, we design an accelerated Schwarz method for radiative transfer equations that makes use of approximate local solution maps constructed offline via a random sampling strategy. Numerical examples demonstrate the accuracy, robustness, and efficiency of the proposed approach. [ABSTRACT FROM AUTHOR]

We study the Schwarz overlapping domain decomposition method applied to the Poisson problem on a special family of domains, which by construction consist of a union of a large number of fixed-size subdomains. These domains are motivated by applications in computational chemistry where the subdomains consist of van der Waals balls. As is usual in the theory of domain decomposition methods, the rate of convergence of the Schwarz method is related to a stable subspace decomposition. We derive such a stable decomposition for this family of domains and analyze how the stability "constant" depends on relevant geometric properties of the domain. For this, we introduce new descriptors that are used to formalize the geometry for the family of domains. We show how, for an increasing number of subdomains, the rate of convergence of the Schwarz method depends on specific local geometry descriptors and on one global geometry descriptor. The analysis also naturally provides lower bounds in terms of the descriptors for the smallest eigenvalue of the Laplace eigenvalue problem for this family of domains. [ABSTRACT FROM AUTHOR]

In this work, we study a local adaptive smoothing algorithm for a-posteriori-steered p-robust multigrid methods. The solver tackles a linear system which is generated by the discretization of a second-order elliptic diffusion problem using conforming finite elements of polynomial order p ≥ 1 {p\geq 1}. After one V-cycle ("full-smoothing" substep) of the solver of [A. Miraçi, J. Papež, and M. Vohralík, A-posteriori-steered p-robust multigrid with optimal step-sizes and adaptive number of smoothing steps, SIAM J. Sci. Comput. 2021, 10.1137/20M1349503], we dispose of a reliable, efficient, and localized estimation of the algebraic error. We use this existing result to develop our new adaptive algorithm: thanks to the information of the estimator and based on a bulk-chasing criterion, cf. [W. Dörfler, A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal. 33 1996, 3, 1106–1124], we mark patches of elements with increased estimated error on all levels. Then, we proceed by a modified and cheaper V-cycle ("adaptive-smoothing" substep), which only applies smoothing in the marked regions. The proposed adaptive multigrid solver picks autonomously and adaptively the optimal step-size per level as in our previous work but also the type of smoothing per level (weighted restricted additive or additive Schwarz) and concentrates smoothing to marked regions with high error. We prove that, under a numerical condition that we verify in the algorithm, each substep (full and adaptive) contracts the error p-robustly, which is confirmed by numerical experiments. Moreover, the proposed algorithm behaves numerically robustly with respect to the number of levels as well as to the diffusion coefficient jump for a uniformly-refined hierarchy of meshes. [ABSTRACT FROM AUTHOR]

In this paper, we have constructed an iterative numerical method based on an overlapping Schwarz procedure with uniform mesh for singularly perturbed fourth-order of convection-diffusion type. The method splits the original domain into two overlapping subdomains. A hybrid difference scheme is proposed in which on the boundary layer region we use the central finite difference scheme on a uniform mesh while on the non-layer region we use the mid-point difference scheme on a uniform mesh. It is shown that the method produces numerical approximations which converge in the maximum norm to the exact solution. We prove that, when appropriate subdomains are used the method produces convergence of almost second-order. Furthermore, it is shown that, two iterations are sufficient to achieve the expected accuracy. Numerical examples are presented to support the theoretical results.

This paper presents a new numerical approach for solving a class of stochastic partial differential systems with additive noise on overlapping subdomains. The domain decomposition method, the deterministic method of lines and the barycentric interpolation method are combined. In addition to these three methods, the stochastic Itô-Taylor family schemes for solving stochastic advection-diffusion-reaction problems is implemented. The solution of the stochastic system is then carried out by collecting interior and interface solutions. Finally, computational results on two dimensional overlapped subdomains with nonlinear boundaries are simulated. [ABSTRACT FROM AUTHOR]

In this work, we consider conforming finite element discretizations of arbitrary polynomial degree p \geq 1 of the Poisson problem. We propose a multilevel a posteriori estimator of the algebraic error. We prove that this estimator is reliable and efficient (represents a two-sided bound of the error), with a constant independent of the degree p. We next design a multilevel iterative algebraic solver from our estimator and show that this solver contracts the algebraic error on each iteration by a factor bounded independently of p. Actually, we show that these two results are equivalent. The p-robustness results rely on the work of Sch\"oberl et al. [IMA J. Numer. Anal., 28 (2008), pp. 1-24] for one given mesh. We combine this with the design of an algebraic residual lifting constructed over a hierarchy of nested unstructured, possibly highly graded, simplicial meshes. The lifting includes a global coarse-level solve with the lowest polynomial degree one together with local contributions from the subsequent mesh levels. These contributions, of the highest polynomial degree p on the finest mesh, are given as solutions of mutually independent local Dirichlet problems posed over overlapping patches of elements around vertices. The construction of this lifting can be seen as one geometric V-cycle multigrid step with zero pre- and one postsmoothing by (damped) additive Schwarz (block Jacobi). One particular feature of our approach is the optimal choice of the step-size generated from the algebraic residual lifting. Numerical tests are presented to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we have constructed an iterative numerical method based on an overlapping Schwarz procedure with uniform mesh for singularly perturbed fourth-order of convection-diffusion type. The method splits the original domain into two overlapping subdomains. A hybrid difference scheme is proposed in which on the boundary layer region we use the central finite difference scheme on a uniform mesh while on the non-layer region we use the mid-point difference scheme on a uniform mesh. It is shown that the method produces numerical approximations which converge in the maximum norm to the exact solution. We prove that, when appropriate subdomains are used the method produces convergence of almost second-order. Furthermore, it is shown that, two iterations are sufficient to achieve the expected accuracy. Numerical examples are presented to support the theoretical results. [ABSTRACT FROM AUTHOR]

When simulating complex physical phenomena such as aircraft icing or de-icing, several dedicated solvers often need to be strongly coupled. In this work, a non-overlapping Schwarz method is constructed with the unsteady simulation of de-icing as the targeted application. To do so, optimized coupling coefficients are first derived for the one dimensional unsteady heat equation with linear boundary conditions and for the steady heat equation with non-linear boundary conditions. The choice of these coefficients is shown to guarantee the convergence of the method. Using a linearization of the boundary conditions, the method is then extended to the case of a general unsteady heat conduction problem. The method is tested on simple cases and the convergence properties are assessed theoretically and numerically. Finally the method is applied to the simulation of an aircraft electrothermal de-icing problem in two dimensions. [ABSTRACT FROM AUTHOR]

Summary: In this article, we have developed an overlapping Schwarz method for a weakly coupled system of convection‐diffusion equations. The method splits the original domain into two overlapping subdomains. A hybrid difference scheme is proposed in which on the boundary layer region, we use the central finite difference scheme on a uniform mesh, whereas on the nonlayer region, we use the mid‐point difference scheme on a uniform mesh. It is shown that the numerical approximations converge in the maximum norm to the exact solution. We have proved that, when appropriate subdomains are used, the method produces almost second‐order convergence. Furthermore, it is shown that two iterations are sufficient to achieve the expected accuracy. Numerical examples are presented to support the theoretical results. The main advantage of this method used with the proposed scheme is that it reduces iteration counts very much and easily identifies in which iteration the Schwarz iterate terminates. [ABSTRACT FROM AUTHOR]

In isogeometric analysis (IGA) the domain of interest is usually represented by B-spline or NURBS patches, as they are present in standard CAD models. Complex domains can often be represented as a union of simple overlapping subdomains, parameterized by (tensor-product) spline patches. Numerical simulation on such overlapping multi-patch domains is a serious challenge in IGA. To obtain non-overlapping subdomains one would usually reparameterize the domain or trim some of the patches. Alternatively, one may use methods that can handle overlapping subdomains. In this paper, we propose a non-iterative, robust and efficient method defined directly on overlapping multi-patch domains. Consequently, the problem is divided into several sub-problems, which are coupled in an appropriate way. The resulting system can be solved directly in a single step. We compare the proposed method with iterative Schwarz domain decomposition approaches and observe that our method reduces the computational cost significantly, especially when handling subdomains with small overlaps. Summing up, our method significantly simplifies the domain parameterization problem, since we can represent any domain of interest as a union of overlapping patches without the need to introduce trimming curves/surfaces. The performance of the proposed method is demonstrated by several numerical experiments for the Poisson problem and linear elasticity in two and three dimensions. • We consider overlapping multi-patch domains without trimming. • We present a non-iterative method to solve PDEs on such domains. • Numerical evidence suggests that the method is robust and efficient. • We compare the method with Schwarz-type methods with respect to stability and timing. • Using the proposed method significantly simplifies the parameterization problem. [ABSTRACT FROM AUTHOR]

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