Discontinuous Solutions Research Articles

Legendre spectral element for plastic localization problems at large scale strains

In paper the method of spectral elements based on the Legendre polynomial for timeindependent elastic-plastic plane problems at large strains is proposed. The method of spectral elements is based on the variational principle (Galerkin’s method). The solution of these problems has the phenomenon of localization of plastic deformations in narrow areas called slip-line or shear band. The possibility of using a spectral element for the numerical solution of these problems with discontinuous solutions is investigated. The yield condition of the material is the von Mises criterion. The stresses are integrated by the radial return method by backward implicit Euler scheme. The system of nonlinear algebraic equations is solved by the Newton’s iterative method. A numerical solution is given of an example of stretching a strip weakened by cuts with a circular base in a plane stress and plane deformed state. Kinematic fields and limit load are obtained. Comparisons of numerical results with the analytical solution obtained for incompressible media constructed by the method of characteristics are presented.

High-Order Conservative Positivity-Preserving DG-Interpolation for Deforming Meshes and Application to Moving Mesh DG Simulation of Radiative Transfer

Solution interpolation between deforming meshes is an important component for several applications in scientific computing, including indirect arbitrary-Lagrangian-Eulerian and rezoning moving mesh methods in numerical solution of partial differential equations. In this paper, a high-order, conservative, and positivity-preserving interpolation scheme is developed based on the discontinuous Galerkin solution of a linear time-dependent equation on deforming meshes. The scheme works for bounded but otherwise arbitrary mesh deformation from the old mesh to the new one. The cost and positivity preservation (with a linear scaling limiter) of the DG-interpolation are investigated. Numerical examples are presented to demonstrate the properties of the interpolation scheme. The DG-interpolation is applied to the rezoning moving mesh DG solution of the radiative transfer equation, an integro-differential equation modeling the conservation of photons and involving time, space, and angular variables. Numerical results obtained for examples in one and two spatial dimensions with various settings show that the resulting rezoning moving mesh DG method maintains the same convergence order as the standard DG method, is more efficient than the method with a fixed uniform mesh, and is able to preserve the positivity of the radiative intensity.

A Discontinuous Galerkin Surface Integral Solution for Scattering From Homogeneous Objects With High Dielectric Constant

The discontinuous Galerkin (DG) method for homogeneous bodies has been studied and shown to be an efficient tool for multiscale homogeneous bodies. However, the slow convergence of DG with the block diagonal preconditioner (BDP) is still observed in solving high contrast homogeneous bodies. An efficient preconditioning approach is designed for the DG method in this communication by using the sparsing approach on the near-field matrix of the whole region. The iteration convergence speed of the DG method is improved while the computing resources for constructing the preconditioner are effectively reduced. Numerical experiments demonstrate the capability of the presented DG method for multiscale homogeneous bodies, especially for those with a high dielectric constant.

Discontinuous finite element solutions for coupled radiation-conduction heat transfer in irregular media

The discontinuous finite element method (DFEM) is used to investigate the coupled radiation-conduction heat transfer in an irregular medium, and the highly accurate solutions for several typical media are numerically obtained. Comparing with the traditional continuous finite element method, the computational domain in the DFEM application is discretized into unstructured meshes that are assumed to be separated from each other. The shape function construction, field variable approximation, and numerical solutions are obtained for every single element. The continuity of the computational domain is maintained by modeling a numerical flux with the up-winding scheme. Thus the DFEM has the salient feature of geometry flexibility and simultaneously supports locally conservative solutions. The DFEM discretization for the radiative transfer equation and the energy diffusion equation are first presented, and the accuracies of the DFEM for coupled radiation-conduction heat transfer problems are verified. Combined radiation-conduction heat transfer problems in several irregular media are afterward solved, and the highly accurate DFEM solutions are presented.

Path-conservative in-cell discontinuous reconstruction schemes for non conservative hyperbolic systems

We are interested in the numerical approximation of discontinuous solutions in non conservative hyperbolic systems. We introduce the basics of a new strategy based on in-cell discontinuous reconstructions to deal with this challenging topic, and apply it to a 2x2 non conservative toy model, and a 3x3 gas dynamics system in Lagrangian coordinates. The strategy allows in particular to compute exactly isolated shocks. Numerical evidences are proposed.

Об устойчивости трубок разрывных решений билинейных систем с запаздыванием

Об устойчивости трубок разрывных решений билинейных систем с запаздыванием

Stieltjes Differential in Impulse Nonlinear Problems

An impulse nonlinear problem admitting discontinuous solutions that are functions of bounded variation is studied. This problem models the deformation of a discontinuous string (chains of strings fastened together by springs) with elastic supports in the form of linear and nonlinear springs (for example, springs with different turns, whose deformations do not obey Hooke’s law). The model is described by a second-order differential equation with derivatives in special measures and Dirichlet boundary conditions. Existence theorems are proved, and conditions for the existence of nonnegative solutions are obtained.

Fine elastic circular inclusion in the area of harmonic vibrations of an unlimited body under smooth contact

Fine elastic circular inclusion in the area of harmonic vibrations of an unlimited body under smooth contact The problem of the interaction of harmonic waves with a thin elastic circular inclusion, which is located in an elastic isotropic body (matrix), is solved. On both sides of the inclusion between it and the body (matrix), the conditions of smooth contact are realized. The solution method is based on representing the displacements in the matrix through discontinuous solutions of the Lamé equations for harmonic vibrations. This made it possible to reduce the problem to Fredholm integral equations of the second kind with respect to functions associated with jumps in normal stress and radial displacement to included ones. After the realization of the boundary conditions on the sides of the inclusion, a system of singular integral equations is obtained to determine these jumps. Keywords: elastic inclusions, cylindrical waves, matrix, stress intensity factor.

Continuity of weak solutions to nonlinear fourth-order equations with strengthened ellipticity via Wolff potentials

In the present article nonlinear fourth-order equations in the divergence form with L^1-right-hand sides and the strengthened ellipticity condition on the coefficients are analyzed. Such equations, but with sufficiently regular right-hand sides, first appeared in the works of Professor I.V. Skrypnik concerning the regularity of generalized solutions for multidimensional nonlinear elliptic equations of high order. This class of equations correctly generalizes the corresponding nonlinear second-order elliptic equations with non-standard growth conditions on the coefficients, which are models for numerous physical phenomena in non-homogeneous medium. The main result of the article is a theorem on an estimation of oscillations in a ball of solutions to the given equations via the Wolff potentials of their right-hand sides. To prove this, we use the improved Kilpelainen-Maly method and pointwise potential estimates of functions related to special subclasses of Sobolev spaces, akin to the well-known De Giorgi classes. A new point is the verification that these classes contain superpositions of solutions and Moser logarithmic functions that include the Wolf potential of the right-hand side of the equation. As a corollary, a new result is obtained on the interior continuity of solutions to the equations with right-hand sides from the Kato class, which is characterized by the uniform convergence to zero of the corresponding Wolff potentials. Some important cases of fulfilling this condition are considered: the right-hand side of the equation belongs to the Morrey space with an index exceeding a certain limiting value, then the solutions are locally Holder continuous; if the right-hand side belongs to the borderline Lorentz-Zygmund classes, then the solutions are only locally continuous, but they are not Holder continuous in the domain. In the case when the summability exponents of the right-hand sides of the equations under consideration are less than the borderline values, there are examples of unbounded discontinuous solutions. These facts are exact analogues of the corresponding results in the theory of second-order elliptic equations.

Solution property preserving reconstruction for finite volume scheme: a boundary variation diminishing+multidimensional optimal order detection framework

SummaryThe purpose of this work is to build a general framework to reconstruct the underlying fields within a finite volume (FV) scheme solving a hyperbolic system of PDEs (Partial Differential Equations). In an FV context, the data are piecewise constants per computational cell and the physical fields are reconstructed taking into account neighbor cell values. These reconstructions are further used to evaluate the physical states usually used to feed a Riemann solver which computes the associated numerical fluxes. The physical field reconstructions must obey some properties linked to the system of PDEs such as the positivity, but also some numerically based ones, like an essentially nonoscillatory behavior. Moreover, the reconstructions should be highly accurate for smooth flows and robust/stable for discontinuous solutions. To ensure a solution property preserving reconstruction, we introduce a methodology to blend high‐/low‐order polynomials and hyperbolic tangent reconstructions. A boundary variation diminishing algorithm is employed to select the best reconstruction in each cell. An a posteriori MOOD detection procedure is employed to ensure the positivity by recomputing the rare problematic cells using a robust first‐order FV scheme. We illustrate the performance of the proposed scheme via the numerical simulations for some benchmark tests which involve vacuum or near vacuum states, strong discontinuities, and also smooth flows. The proposed scheme maintains high accuracy on smooth profile, preserves the positivity and can eliminate the oscillations in the vicinity of discontinuities while maintaining sharper discontinuity with superior solution quality compared to classical highly accurate FV schemes.

A modified sensitivity equation method for the Euler equations in presence of shocks

AbstractThe continuous sensitivity equation method allows to quantify how changes in the input of a partial differential equation (PDE) model affect the outputs, by solving additional PDEs obtained by differentiating the model. However, this method cannot be used directly in the framework of hyperbolic PDE systems with discontinuous solution, because it yields Dirac delta functions in the sensitivity solution at the location of state discontinuities. This difficulty is well known from theoretical viewpoint, but only a few works can be found in the literature regarding the possible numerical treatment. Therefore, we investigate in this study how classical numerical schemes for compressible Euler equations can be modified to account for shocks when computing the sensitivity solution. In particular, we propose the introduction of a source term, that allows to remove the spikes associated to the Dirac delta functions in the numerical solution. Numerical studies exhibit a strong impact of the numerical diffusion on the accuracy of this strategy. Therefore, we propose an anti‐diffusive numerical scheme coupled with the approximate Riemann solver of Roe for the state problem. For the sensitivity problem, two different numerical schemes are implemented and compared: one which takes into account the contact wave and another that neglects it. The effects of the numerical diffusion on the convergence of the schemes with respect to the grid are discussed. Finally, an application to uncertainty propagation is investigated and the different numerical schemes are compared.

Strength calculation of risers near the supports of reinforced concrete three-hinged frames based on the concrete plasticity theory

The necessity of estimation of residual strength of risers sections near supports of reinforced concrete three-hinged frames at their reuse after operation in aggressive environment conditions and damages presence is indicated. The method of calculating the elements strength under the shear on the joint action of the transverse and longitudinal forces based on the plasticity theory is elaborated. The variational method, the virtual velocities principle, discontinuous solutions are applied. The kinematically possible failure schemes, the realization of which depends on the values ratio of transverse and longitudinal forces are proposed. The application boundary of these schemes is defined. The permissible level of concrete class reduction and the degree of its integrity violation were established, under which the residual strength of serial three-hinged frames near the supports was provided.

Open-boundary spectral and flux-balance Vlasov simulation

Simulations of one-dimensional Vlasov–Maxwell solutions with non-periodic boundary conditions are discussed. Results obtained using a recently developed filtered flux-balance simulation system are compared to those obtained using a filtered, Fourier–Fourier transformed system. Excellent agreement is confirmed except for the appearance of the Gibbs phenomenon on the discontinuous simulated solutions of the transformed system. Recovery of the flux-balance results from the Fourier transformed results using the inverse polynomial reconstruction method is demonstrated.

A locally second order symmetric method for discontinuous solution of Poisson's equation on uniform cartesian grids

A new method is proposed for numerically solving the Poisson equation for non-continuous scalar fields on a uniform Cartesian grid. The sharp discontinuity in both the magnitude and the gradient of the scalar field normal to the interface is represented by the numerical solution with second order accuracy at the interface. This is achieved by setting up a composite solution, which is a weighted average of two fictitious scalar fields that together produce the required discontinuity within each interfacial grid cell. A smooth treatment of the Poisson coefficient in a narrow band around the interface allows sharp interfacial jumps to be expressed with second order accuracy on regular grid points around the interface using a standard signed distance function. Moreover, the jump in the gradient tangent to the interface is not needed to enforce the jump in the gradient normal to the interface. The resulting linear system is symmetric and leads to second order accurate solutions on grid points adjacent to the interface. The accuracy of the new framework is compared with other methods.

Well-Balanced Discontinuous Galerkin Method for Shallow Water Equations with Constant Subtraction Techniques on Unstructured Meshes

The classical Saint–Venant shallow water equations on complex geometries have wide applications in many areas including coastal engineering and atmospheric modeling. The main numerical challenge in simulating Saint–Venant equations is to maintain the high order of accuracy and well-balanced property simultaneously. In this paper, we propose a high-order accurate and well-balanced discontinuous Galerkin (DG) method on two dimensional (2D) unstructured meshes for the Saint–Venant shallow water equations. The technique used to maintain well-balanced property is called constant subtraction and proposed in Yang et al. (J Sci Comput 63:678–698, 2015). Hierarchical reconstruction limiter with a remainder correction technique is introduced to control numerical oscillations. Numerical examples with smooth and discontinuous solutions are provided to demonstrate the performance of our proposed DG methods.

On a compact finite-difference scheme of the third order of weak approximation

The stability and accuracy of compact difference schemes with artificial viscosities of the fourth divergence order are studied. These schemes have a third order both of classical approximation on smooth solutions and weak approximation on discontinuous solutions. As a result of the stability analysis of these schemes in the linear approximation, the optimal values of their viscosity coefficients were obtained. Test calculations are presented to demonstrate the advantages of the new compact scheme compared to the TVD and WENO schemes when calculating discontinuous solutions with shock waves.

Structural stability of shock waves in 2D compressible elastodynamics

We study the two-dimensional structural stability of shock waves in a compressible isentropic inviscid elastic material in the sense of the local-in-time existence and uniqueness of discontinuous shock front solutions of the equations of compressible elastodynamics in two space dimensions. By the energy method based on a symmetrization of the wave equation and giving an a priori estimate without loss of derivatives for solutions of the constant coefficients linearized problem we find a condition sufficient for the uniform stability of rectilinear shock waves. Comparing this condition with that for the uniform stability of shock waves in isentropic gas dynamics, we make the conclusion that the elastic force plays stabilizing role. In particular, we show that, as in isentropic gas dynamics, all compressive shock waves are uniformly stable for convex equations of state. Moreover, for some particular deformations (and general equations of state), by the direct test of the uniform Kreiss–Lopatinski condition we show that the stability condition found by the energy method is not only sufficient but also necessary for uniform stability. As is known, uniform stability implies structural stability of corresponding curved shock waves.

A conservative solver for surface-tension-driven multiphase flows on collocated unstructured grids

We developed a novel conservative model for simulating incompressible multiphase flows on collocated unstructured grids. In conventional schemes, the divergence-free condition is enforced by the numerical approximations based on the least-square approach which inherently cannot guarantee its conservative property for volume integrated average (VIA) of velocity. In this study, point values (PVs) at cell vertices are defined in addition to its VIAs for the pressure variable and a novel discretized formulation is devised for the pressure Poisson equation so as to enforce the VIA of velocity to satisfy the divergence-free condition. Considering the large density ratios in the vicinity of fluid interface, we have also put forward a non-oscillatory and less dissipative reconstruction scheme to improve the resolvability of discontinuous solutions. For the surface-tension dominated flow problems, a novel reinitialization scheme is proposed to transform the abruptly-varying volume fractions into a smooth-varying level-set function which significantly improves the solution accuracy of curvature estimation so as to suppress the spurious currents in presence of the unphysical oscillation. The resulting multiphase model that combines above numerical methods and techniques, therefore adequately assures divergence-free condition to preserve mass conservation, effectively controls the numerical oscillations and dissipations in the vicinity of moving interface and accurately resolves surface tension force with substantially suppressed spurious currents. Various numerical examples have been presented which demonstrate the excellent performance of the newly developed model to predict both fluid dynamics and interfacial deformations with high-fidelity.

{h-p-hp}-Multilevel discontinuous Galerkin solution strategies for elliptic operators

In this work, we investigate the performance of -multilevel preconditioners for discontinuous Galerkin (dG) discretisations of elliptic operators with constant coefficients. Recent publications targeting multilevel solution strategies for incompressible fluid flow computations demonstrated that dG discretisation of viscous terms require ad hoc inherited multilevel preconditioners. Accordingly, we consider elliptic operators discretised by means of the BR2 dG method introduced by Bassi and Rebay and we compare agglomeration-based hp-coarsening with previously introduced -multilevel strategies. The numerical results show that, when the polynomial degree is sufficiently high and the mesh is sufficiently dense, the hp-multilevel preconditioner can be fruitfully exploited.

Investigation of an Idealized Virus Capsid Model by the Dynamic Elasticity Apparatus

The three-dimensional dynamic theory of elasticity is applied to investigate the mechanical properties of the virus capsid. An idealized model of viruses is based on the 3D boundary-value problem of mathematical physics formulated in a spherical coordinate system for the steady-state oscillation process. The virus is modeled by a hollow elastic sphere filled with an acoustic medium and located in a different acoustic medium. The stated boundary-value problem is solved with the help of the method of integral transforms and the method of the discontinuous solutions. As a result, the exact solution of the problem is obtained. The numerical calculations of the elastic characteristics of the virus are carried out.