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Abstract: This paper is concerned with mathematical and numerical analysis of the system of radiative integral transfer equations. The existence and uniqueness of solution to the integral system is proved by establishing the boundedness of the radiative integral operators and proving the invertibility of the operator matrix associated with the system. A collocation-boundary element method is developed to discretize the differential-integral system. For the non-convex geometries, an element-subdivision algorithm is developed to handle the computation of the integrals containing the visibility factor. An efficient iterative algorithm is proposed to solve the nonlinear discrete system and its convergence is also established. Numerical experiment results are also presented to verify the effectiveness and accuracy of the proposed method and algorithm. PubDate: 2019-02-01 DOI: 10.21136/AM.2019.0276-17
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Abstract: Unique solvability and stability analysis is conducted for a generalized particle method for a Poisson equation with a source term given in divergence form. The general- ized particle method is a numerical method for partial differential equations categorized into meshfree particle methods and generally indicates conventional particle methods such as smoothed particle hydrodynamics and moving particle semi-implicit methods. Unique solv- ability is derived for the generalized particle method for the Poisson equation by introducing a connectivity condition for particle distributions. Moreover, stability is obtained for the discretized Poisson equation by introducing discrete Sobolev norms and a semi-regularity condition of a family of discrete parameters. PubDate: 2019-02-01 DOI: 10.21136/AM.2019.0210-18
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Abstract: This paper introduces the notion of pairwise and coordinatewise negative dependence for random vectors in Hilbert spaces. Besides giving some classical inequalities, almost sure convergence and complete convergence theorems are established. Some limit theorems are extended to pairwise and coordinatewise negatively dependent random vectors taking values in Hilbert spaces. An illustrative example is also provided. PubDate: 2019-02-01 DOI: 10.21136/AM.2018.0060-18
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Abstract: A 0/1-simplex is the convex hull of n+1 affinely independent vertices of the unit n-cube In. It is nonobtuse if none of its dihedral angles is obtuse, and acute if additionally none of them is right. Acute 0/1-simplices in In can be represented by 0/1-matrices P of size n × n whose Gramians G = P⊤P have an inverse that is strictly diagonally dominant, with negative off-diagonal entries. In this paper, we will prove that the positive part D of the transposed inverse P−⊤ of P is doubly stochastic and has the same support as P. In fact, P has a fully indecomposable doubly stochastic pattern. The negative part C of P−⊤ is strictly row-substochastic and its support is complementary to that of D, showing that P−⊤ = D−C has no zero entries and has positive row sums. As a consequence, for each facet F of an acute 0/1-facet S there exists at most one other acute 0/1-simplex Ŝ in In having F as a facet. We call Ŝ the acute neighbor of S at F. If P represents a 0/1-simplex that is merely nonobtuse, the inverse of G = P⊤P is only weakly diagonally dominant and has nonpositive off-diagonal entries. These matrices play an important role in finite element approximation of elliptic and parabolic problems, since they guarantee discrete maximum and comparison principles. Consequently, P−⊤ can have entries equal to zero. We show that its positive part D is still doubly stochastic, but its support may be strictly contained in the support of P. This allows P to have no doubly stochastic pattern and to be partly decomposable. In theory, this might cause a nonobtuse 0/1-simplex S to have several nonobtuse neighbors Ŝ at each of its facets. In this paper, we study nonobtuse 0/1-simplices S having a partly decomposable matrix representation P. We prove that if S has such a matrix representation, it also has a block diagonal matrix representation with at least two diagonal blocks. Moreover, all matrix representations of S will then be partly decomposable. This proves that the combinatorial property of having a fully indecomposable matrix representation with doubly stochastic pattern is a geometrical property of a subclass of nonobtuse 0/1-simplices, invariant under all n-cube symmetries. We will show that a nonobtuse simplex with partly decomposable matrix representation can be split in mutually orthogonal simplicial facets whose dimensions add up to n, and in which each facet has a fully indecomposable matrix representation. Using this insight, we are able to extend the one neighbor theorem for acute simplices to a larger class of nonobtuse simplices. PubDate: 2019-02-01 DOI: 10.21136/AM.2018.0207-18
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Abstract: The dynamics of an activator-inhibitor model with general cubic polynomial source is investigated. Without diffusion, we consider the existence, stability and bifurcations of equilibria by both eigenvalue analysis and numerical methods. For the reaction-diffusion system, a Lyapunov functional is proposed to declare the global stability of constant steady states, moreover, the condition related to the activator source leading to Turing instability is obtained in the paper. In addition, taking the production rate of the activator as the bifurcation parameter, we show the decisive effect of each part in the source term on the patterns and the evolutionary process among stripes, spots and mazes. Finally, it is illustrated that weakly linear coupling in the activator-inhibitor model can cause synchronous and anti-phase patterns. PubDate: 2019-02-01 DOI: 10.21136/AM.2019.0142-18
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Abstract: A variational two-level method in the class of methods with an aggressive coarsening and a massive polynomial smoothing is proposed. The method is a modification of the method of Section 5 of Tezaur, Vaněk (2018). Compared to that method, a significantly sharper estimate is proved while requiring only slightly more computational work. PubDate: 2018-12-01 DOI: 10.21136/AM.2018.0314-17
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Abstract: The paper is concerned with the application of the space-time discontinuous Galerkin method (STDGM) to the numerical solution of the interaction of a compressible flow and an elastic structure. The flow is described by the system of compressible Navier-Stokes equations written in the conservative form. They are coupled with the dynamic elasticity system of equations describing the deformation of the elastic body, induced by the aerodynamical force on the interface between the gas and the elastic structure. The domain occupied by the fluid depends on time. It is taken into account in the Navier-Stokes equations rewritten with the aid of the arbitrary Lagrangian-Eulerian (ALE) method. The resulting coupled system is discretized by the STDGM using piecewise polynomial approximations of the sought solution both in space and time. The developed method can be applied to the solution of the compressible flow for a wide range of Mach numbers and Reynolds numbers. For the simulation of elastic deformations two models are used: the linear elasticity model and the nonlinear neo-Hookean model. The main goal is to show the robustness and applicability of the method to the simulation of the air flow in a simplified model of human vocal tract and the flow induced vocal folds vibrations. It will also be shown that in this case the linear elasticity model is not adequate and it is necessary to apply the nonlinear model. PubDate: 2018-12-01 DOI: 10.21136/AM.2018.0139-18
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Abstract: In specific fields of research such as preservation of historical buildings, medical imaging, geophysics and others, it is of particular interest to perform only a non-intrusive boundary measurements. The idea is to obtain comprehensive information about the material properties inside the considered domain while keeping the test sample intact. This paper is focused on such problems, i.e. synthesizing a physical model of interest with a boundary inverse value technique. The forward model is represented here by time dependent heat equation with transport parameters that are subsequently identified using a modified Calderón problem which is numerically solved by a regularized Gauss-Newton method. The proposed model setup is computationally verified for various domains, loading conditions and material distributions. PubDate: 2018-12-01 DOI: 10.21136/AM.2018.0323-17
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Abstract: The paper deals with formulation and numerical solution of problems of identification of material parameters for continuum mechanics problems in domains with heterogeneous microstructure. Due to a restricted number of measurements of quantities related to physical processes, we assume additional information about the microstructure geometry provided by CT scan or similar analysis. The inverse problems use output least squares cost functionals with values obtained from averages of state problem quantities over parts of the boundary and Tikhonov regularization. To include uncertainties in observed values, Bayesian inversion is also considered in order to obtain a statistical description of unknown material parameters from sampling provided by the Metropolis-Hastings algorithm accelerated by using the stochastic Galerkin method. The connection between Bayesian inversion and Tikhonov regularization and advantages of each approach are also discussed. PubDate: 2018-12-01 DOI: 10.21136/AM.2018.0195-18
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Abstract: The finite element (FE) solution of geotechnical elasticity problems leads to the solution of a large system of linear equations. For solving the system, we use the preconditioned conjugate gradient (PCG) method with two-level additive Schwarz preconditioner. The preconditioning is realised in parallel. A coarse space is usually constructed using an aggregation technique. If the finite element spaces for coarse and fine problems on structural grids are fully compatible, relations between elements of matrices of the coarse and fine problems can be derived. By generalization of these formulae, we obtain an overlapping aggregation technique for the construction of a coarse space with smoothed basis functions. The numerical tests are presented at the end of the paper. PubDate: 2018-12-01 DOI: 10.21136/AM.2018.0142-17
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Abstract: Recent developments in the field of stochastic mechanics and particularly regarding the stochastic finite element method allow to model uncertain behaviours for more complex engineering structures. In reliability analysis, polynomial chaos expansion is a useful tool because it helps to avoid thousands of time-consuming finite element model simulations for structures with uncertain parameters. The aim of this paper is to review and compare available techniques for both the construction of polynomial chaos and its use in computing failure probability. In particular, we compare results for the stochastic Galerkin method, stochastic collocation, and the regression method based on Latin hypercube sampling with predictions obtained by crude Monte Carlo sampling. As an illustrative engineering example, we consider a simple frame structure with uncertain parameters in loading and geometry with prescribed distributions defined by realistic histograms. PubDate: 2018-12-01 DOI: 10.21136/AM.2018.0335-17
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Abstract: In this contribution, we present the problem of shape optimization of the plunger cooling which comes from the forming process in the glass industry. We look for a shape of the inner surface of the insulation barrier located in the plunger cavity so as to achieve a constant predetermined temperature on the outward surface of the plunger. A rotationally symmetric system, composed of the mould, the glass piece, the plunger, the insulation barrier and the plunger cavity, is considered. The state problem is given as a multiphysics problem where solidifying molten glass is cooled from the inside by water flowing through the plunger cavity and from the outside by the environment surrounding the mould. The cost functional is defined as the squared \(L^2_r\) norm of the difference between a prescribed constant and the temperature on the outward boundary of the plunger. The temperature distribution is controlled by changing the insulation barrier wall thickness. The numerical results of the optimization to the required target temperature 800 ◦C of the outward plunger surface together with the distribution of temperatures along the interface between the plunger and the glass piece before, during and after the optimization process are presented. PubDate: 2018-12-01 DOI: 10.21136/AM.2018.0247-17
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Abstract: The paper is devoted to the study of the existence of solutions for nonlinear nonmonotone evolution equations in Banach spaces involving anti-periodic boundary conditions. Our approach in this study relies on the theory of monotone and maximal monotone operators combined with the Schaefer fixed-point theorem and the monotonicity method. We apply our abstract results in order to solve a diffusion equation of Kirchhoff type involving the Dirichlet p-Laplace operator. PubDate: 2018-10-01 DOI: 10.21136/AM.2018.0136-18
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Abstract: This paper is devoted to the study of the linear parabolic problem \(\varepsilon\partial_t{u_\varepsilon}(x,t)-\nabla\cdot(a(x/\varepsilon, t/\varepsilon^3)\nabla{u_\varepsilon}(x,t))=f(x,t)\) by means of periodic homogenization. Two interesting phenomena arise as a result of the appearance of the coefficient ε in front of the time derivative. First, we have an elliptic homogenized problem although the problem studied is parabolic. Secondly, we get a parabolic local problem even though the problem has a different relation between the spatial and temporal scales than those normally giving rise to parabolic local problems. To be able to establish the homogenization result, adapting to the problem we state and prove compactness results for the evolution setting of multiscale and very weak multiscale convergence. In particular, assumptions on the sequence {uε} different from the standard setting are used, which means that these results are also of independent interest. PubDate: 2018-10-01 DOI: 10.21136/AM.2018.0350-17
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Abstract: The paper deals with two mathematical models of predator-prey type where a transmissible disease spreads among the predator species only. The proposed models are analyzed and compared in order to assess the influence of hidden and explicit alternative resource for predator. The analysis shows boundedness as well as local stability and transcritical bifurcations for equilibria of systems. Numerical simulations support our theoretical analysis. PubDate: 2018-10-01 DOI: 10.21136/AM.2018.0158-18
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Abstract: A new hybrid of block-pulse functions and Boubaker polynomials is constructed to solve the inequality constrained fractional optimal control problems (FOCPs) with quadratic performance index and fractional variational problems (FVPs). First, the general formulation of the Riemann-Liouville integral operator for Boubaker hybrid function is presented for the first time. Then it is applied to reduce the problems to optimization problems, which can be solved by the existing method. In this way we find the extremum value of FOCPs without adding slack variables to inequality trajectories. Also we show that if the number of bases is increased, the used approximations in this method are convergent. The applicability and validity of the method are shown by numerical results of some examples, moreover, a comparison with the existing results shows the preference of this method. PubDate: 2018-10-01 DOI: 10.21136/AM.2018.0083-18
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Abstract: We shall prove a weak comparison principle for quasilinear elliptic operators −div(a(x,∇u)) that includes the negative p-Laplace operator, where a: × ℝN → ℝN satisfies certain conditions frequently seen in the research of quasilinear elliptic operators. In our result, it is characteristic that functions which are compared belong to different spaces. PubDate: 2018-08-01 DOI: 10.21136/AM.2018.0126-18
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Authors:Kebaili Zahira; Benterki Djamel Pages: 1 - 16 Abstract: We propose a penalty approach for a box constrained variational inequality problem (BVIP). This problem is replaced by a sequence of nonlinear equations containing a penalty term. We show that if the penalty parameter tends to infinity, the solution of this sequence converges to that of BVIP when the function F involved is continuous and strongly monotone and the box C contains the origin. We develop the algorithmic aspect with theoretical arguments properly established. The numerical results tested on some examples are satisfactory and confirm the theoretical approach. PubDate: 2018-05-22 DOI: 10.21136/am.2018.0334-17
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