The fictitious domain method

Flow simulations in rotary volumetric pumps and compressors with the fictitious domain method

On the application of immersed boundary, fictitious domain and body-conformal mesh methods to many particle multiphase flows

Two closely-related fictitious domain methods for solving problems involving multiple interfaces are introduced. Like other fictitious domain methods, the proposed methods simplify the task of finite element mesh generation and provide access to solvers that can take advantage of uniform structured grids. The proposed methods do not involve the Lagrange multipliers, which makes them quite different from existing fictitious domain methods. This difference leads to an advantageous form of the inf–sup condition, and allows one to avoid time-consuming integration over curvilinear surfaces. In principle, the proposed methods have the same rate of convergence as existing fictitious domain methods. Nevertheless it is shown that, at the cost of introducing additional unknowns, one can improve the quality of the solution near the interfaces. The methods are presented using a two-dimensional model problem formulated in the context of linearized theory of elasticity. The model problem is sufficient for presenting method details and mathematical foundations. Although the model problem is formulated in two dimensions and involves only one interface, there are no apparent conceptual difficulties to extending the methods to three dimensions and multiple interfaces. Further, it is possible to extend the methods to nonlinear problems involving multiple interfaces.

The article proposes original solvability conditions and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem. And we use the matrix pseudo-inversion technique of Moore-Penrose. The posed problem in the article continues the study of conditions of solvability and schemes for finding solutions of the nonlinear Noetherian boundary-value problems given in the monographs by A. Poincare, A.M. Lyapunov, I.G. Malkin, J. Hale, Yu.A. Ryabov, A.M. Samoylenko, N.V. Azbelev, V.P. Maksimov, L.F. Rakhmatullina and A.A. Boychuk. We studied a general case, when a linear bounded operator corresponding to the homogeneous part of the linear Noetherian differential-algebraic boundary value problem has no inverse. Sufficient conditions for reducibility of the differential algebraic equation to the system uniting a differential and algebraic equation are found. Thus, the differential-algebraic boundary value problem is reduced to the nonlinear Noetherian boundary value problem for the system of ordinary differential equations. We studied the case of the presence of simple roots of the equation for generating amplitudes. Constructive necessary and sufficient conditions of existence were obtained to find solutions to the problem in the critical case, and the converging iterative scheme was constructed. The proposed solvability conditions, and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem are illustrated in detail by the example from the nonlinear Noetherian differential-algebraic boundary value problem for Duffing type equations. For control of the rate of the iterative scheme convergence to the exact solution of the differential-algebraic boundary value problem for the Duffing type equation, we used the residuals of the obtained approximations in the Duffing type equation in the space of continuous functions.

The purpose of this paper is to analyze the fluid and moving boundary problem by fictitious domain method with Lagrange multiplier. The incompressible Navier-Stokes equation is used in this paper. To solve this equation, fictitious domain method with Lagrange multiplier is applied. For the analysis of the finite element method, the mixed interpolation based on the bubble function is presented. In fictitious domain method, this method is not necessary to remeshing. Additionally, the calculation time is decreased, and the amount of memory is reduced. It is necessary to approximate by using interpolation function for the Lagrange multiplier. However, it is dicult to integrate the formula. The Delta function of Dirac is used here. Two kinds of meshes are used for fictitious domain method. One of meshes is used as computational domain, the other is used as a bounded domain. In this research, it is analyzed to move the object by the fluid force (drag force) in 3-dimension. In fictitious domain method, it is easy to obtain the fluid force. Because it is defined the Lagrange multiplier is equal to the traction in each node on a bounded domain. The pressure and flow velocity when the object moves are verified. And advantages of fictitious domain method based on distributed Lagrange multiplier are investigated.

Solution of nonlinear boundary value problems — I method of third order convergence for solution of nonlinear two-point boundary value problems single second order equation

Fictitious domain method (FDM) is a commonly accepted direct numerical simulation technique for moving boundary problems. Indicator function used to distinguish the solid zone and the fluid zone is an essential part concerning the whole prediction accuracy of FDM. In this work, a new indicator function through volume intersection is developed for FDM. In this method, the arbitrarily polyhedral cells across the interface between fluid and solid are located and subdivided into tetrahedrons. The fraction of the solid volume in each cell is accurately computed to achieve high precision of integration calculation in the particle domain, improving the accuracy of the whole method. The quadrature over the solid domain shows that the newly developed indicator function can provide results with high accuracy for variable integration in both stationary and moving boundary problems. Several numerical tests, including flow around a circular cylinder, a single sphere in a creeping shear flow, settlement of a circular particle in a closed container, and in-line oscillation of a circular cylinder, have been performed. The results show good accuracy and feasibility in dealing with the stationary boundary problem as well as the moving boundary problem. This method is accurate and conservative, which can be a feasible tool for studying problems with moving boundaries.

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

A novel spectral procedure for the numerical solution of non-linear boundary value problems is presented. The discrete spectral equations are solved by an iterative algorithm using space-domain preconditioning. The preconditioning operator is obtained by spatial weighting (or windowing) of the exact differential operator. Convergence behaviour of the iterative solution is investigated using an eigenvalue analysis. Theoretical estimates for the convergence rate and accuracy are compared with that achieved in a numerical application—a non-linear boundary value problem from semiconductor device modelling. The method combines the infinite-order exponential accuracy of spectral discretizations with the sparse structure of finite difference equations. This offers numerical advantages in comparison to previously developed Fourier–Galerkin algorithms, particularly important for physically ill-conditioned and strongly non-linear problems. The tradeoff of achievable accuracy versus computer time is easily controlled, thus making essential speed-ups possible for moderate accuracy requirements.

The main objective of this paper is to develop new spectral solutions for linear and nonlinear fifth-order boundary value problems. For this aim, a new operational matrix of derivatives of certain nonsymmetric generalized Jacobi polynomials is established. The key idea for obtaining the suggested spectral solutions is to convert the fifth-order linear and nonlinear boundary value problems into systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable solvers. A theoretical error bound is derived and the convergence analysis of the proposed expansion is proved. Numerical tests are presented aiming to ascertain the high efficiency and accuracy of the suggested algorithms.

We analyze the relationship between boundary-value problems with impulsive action at fixed points of time and boundary-value problems with switchings at fixed and nonfixed points of time. We find the constructive conditions of solvability and a scheme for construction of solutions of a nonlinear periodic boundary-value problem with switchings at nonfixed points of time. By using the Adomian decomposition method, we obtain the solvability conditions and construct a new iterative technique for finding solutions of a weakly nonlinear periodic boundary-value problem with switchings at nonfixed points of time. In addition, we obtain constructive conditions for convergence of the iterative scheme to the solution of the weakly nonlinear boundary-value problem as well as the switchings points. The obtained iterative scheme is applied to find approximations to the periodic solution of the equation with switchings at nonfixed points of time, which models a nonisothermal chemical reaction.

Fictitious domain methods facilitate the discretization of boundary value problems by applying simple meshes containing finite elements or cells that do not conform to the geometry of the domain of interest. In this way, the effort of meshing complex domains is shifted to the numerical integration of those elements/cells that are cut by the boundary of the domain. In this chapter, we will first introduce a high-order fictitious domain method and then present adaptive methods that are suited for the numerical integration of broken elements and cells. Since the quadrature schemes presented in this chapter are quite general, they can be applied to the different versions of fictitious domain methods.

Article Solution of the Stokes nonstationary problems by the fictitious domain method was published on January 1, 1995 in the journal Russian Journal of Numerical Analysis and Mathematical Modelling (volume 10, issue 3).

Standard finite element methods employ an element‐wise assembly strategy. The element's contribution to the system matrix is formed by a loop over quadrature points. This concept is also used in fictitious domain methods, which perform simulations on a simple tensor‐product background mesh cut by a boundary representation that defines the domain of interest.Considering such d‐dimensional background meshes based on splines of degree p with maximal smoothness, Cp−1, the cost of setting up the system matrix is 𝒪(p3d) per degree of freedom. Alternative assembly and formation techniques can significantly reduce this cost. In particular, the combination of (1) sum factorization, (2) weighted quadrature, and (3) row‐based assembly yields a cost of 𝒪(pd+1) for non‐cut background meshes. However, applying this fast approach to cut background meshes is an open challenge since they do not have a tensor‐product structure.This work presents techniques that allow the treatment of cut background meshes and thus the application of fast formation and assembly to fictitious domain methods. First, a discontinuous version of weighted quadrature is presented, which introduces a discontinuity into a cut test function's support. The cut region can be treated separately from the non‐cut counterpart; the latter can be assembled by the fast concepts. A three‐dimensional example investigates the accuracy and efficiency of the proposed concept and demonstrates its speed‐up compared to conventional formation and assembly.

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