A hybrid formulation of eddy current problems

In this article we examine the well‐known magneto‐quasistatic eddy current model for the behavior of low‐frequency electromagnetic fields. We restrict ourselves to formulations in the frequency domain and linear materials, but admit rather general topological arrangements. The generic eddy current model allows two dual formulations, which may be dubbed E‐based and H‐based. We investigate the so‐called hybrid approach that combines both formulations by means of coupling conditions across the boundaries of conducting regions. The resulting continuous and discrete variational formulations will be discussed, and an optimal error estimate for edge finite elements will be proved. It is worthy to note that for this approach no difficulties arise from the topology of the conducting regions. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

A hybrid coupling interface method for elliptic complex interface problems

We propose a coupling interface method (CIM) under Cartesian grid for solving elliptic complex interface problems in arbitrary d dimensions, where the coefficients, the source terms, and the solutions may be discontinuous or singular across the interfaces. It consists of a first-order version (CIM1) and a second-order version (CIM2). In one dimension, this finite difference method at a grid point adjacent to the interface is derived based on piecewise linear (CIM1) or quadratic (CIM2) approximation of the solution and two jump conditions. The method is extended to high dimensions through a dimensionby-dimension approach. To connect information from each dimension, a coupled equation for the principal derivatives is derived through the jump conditions in each coordinate direction. For CIM2, one-side interpolation for cross derivatives is need. This coupling approach reduces number of grid point in the finite difference stencil. The hybrid method uses CIM1 or CIM2 adaptly for complex interface. Numerical tests demonstrate that CIM1 and CIM2 are respectively first order and second order in the maximal norm with less error as compared with other methods. In addition, the hybrid CIM can solve complex interface problems in two and three dimensions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A level-set method for shape optimization

We study a level-set method for numerical shape optimization of elastic structures. Our approach combines the level-set algorithm of Osher and Sethian with the classical shape gradient. Although this method is not specifically designed for topology optimization, it can easily handle topology changes for a very large class of objective functions. Its cost is moderate since the shape is captured on a fixed Eulerian mesh. To cite this article: G. Allaire et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1125–1130.     

An interface-fitted adaptive mesh method for elliptic problems and its application in free interface problems with surface tension

This work presents a novel two-dimensional interface-fitted adaptive mesh method to solve elliptic problems of jump conditions across the interface, and its application in free interface problems with surface tension. The interface-fitted mesh is achieved by two operations: (i) the projection of mesh nodes onto the interface and (ii) the insertion of mesh nodes right on the interface. The interface-fitting technique is combined with an existing adaptive mesh approach which uses addition/subtraction and displacement of mesh nodes. We develop a simple piecewise linear finite element method built on this interface-fitted mesh and prove its almost optimal convergence for elliptic problems with jump conditions across the interface. Applications to two free interface problems, a sheared drop in Stokes flow and the growth of a solid tumor, are presented. In these applications, the interface surface tension serves as the jump condition or the Dirichlet boundary condition of the pressure, and the pressure is solved with the interface-fitted finite element method developed in this work. In this study, a level-set function is used to capture the evolution of the interface and provide the interface location for the interface fitting.     

Lyapunov-barrier characterization of robust reach–avoid–stay specifications for hybrid systems

Stability, reachability, and safety are crucial properties of dynamical systems. While verification and control synthesis of reach–avoid–stay objectives can be effectively handled by abstraction-based formal methods, such approaches can be computationally expensive due to the use of state–space discretization. In contrast, Lyapunov methods qualitatively characterize stability and safety properties without any state–space discretization. Recent work on converse Lyapunov-barrier theorems also demonstrates an approximate completeness for verifying reach–avoid–stay specifications of systems modeled by nonlinear differential equations. In this paper, based on the topology of hybrid arcs, we extend the Lyapunov-barrier characterization to more general hybrid systems described by differential and difference inclusions. We show that Lyapunov-barrier functions are not only sufficient to guarantee reach–avoid–stay specifications for well-posed hybrid systems, but also necessary for arbitrarily slightly perturbed systems under mild conditions. Numerical examples are provided to illustrate the main results.     

New methods in fluid-structure coupling with application to biomechanics

An Eulerian approach is presented for generic fluid-structure coupling of an elastic body with an incompressible fluid. We consider the coupling as a multiphysics problem where fluid-solid interfaces are captured by a level-set method. The main features of the method are its simplicity, and its natural control of mass and energy. We are indeed able to prove an energy equation which ensures in particular that the regularization of the force does not involve any energy dissipation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fluid-Structure Interactions in ALE and Fully Eulerian Coordinates

We present a novel approach to treat fluid-structure interactions in a closed variational setting. The standard model is the so-called ‘arbitrary Lagrangian-Eulerian’ (ALE) framework. Here, the fluid is transformed into an artificial coordinate system, which fits with the Lagrangian structure system. In the fully Eulerian framework, which is the reverse to ALE, the fluid is left in its natural coordinates while structure is transformed. With this approach, very large deformations, change of topology as well as free movement of the structure can be computed. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A numerical method based on hybrid orthonormal Bernstein and improved block-pulse functions for solving Volterra–Fredholm integral equations

This paper deals with the numerical solution of the integral equations of linear second kind Volterra–Fredholm. These integral equations are commonly used in engineering and mathematical physics to solve many of the problems. A hybrid of Bernstein and improved block-pulse functions method is introduced and used where the key point is to transform linear second-type Volterra–Fredholm integral equations into an algebraic equation structure that can be solved using classical methods. Numeric examples are given which demonstrate the related features of the process.     

Eigenpath analysis of rotating mechanical systems based on ALE formulation

When bodies rotate they are subjected to velocity-dependent gyroscopic and inertia effects, which alter the dynamic behaviour of the structure. Analysing its eigenpairs consisting of eigenvalues and eigenvectors, these parameter-dependent changes can be tracked as eigenpaths by means of a continuation algorithm as in [1]. For rotationally symmetric structures an ALE approach (ALE = Arbitrary LAGRANGIAN-EULERIAN) for the finite element method as described in [2] can be employed. It allows simulating rotational influences on the body without actually having to rotate the mesh of the structure itself. This still-standing mesh facilitates further steps like coupling of other, static components with the rotating body. The combination of ALE with eigenpath analysis including an example is described in the following. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coupling of compressible and incompressible flow regions using the multiple pressure variables approach

In many cases, multiphase flows are simulated on the basis of the incompressible Navier–Stokes equations. This assumption is valid as long as the density changes in the gas phase can be neglected. Yet, for certain technical applications such as fuel injection, this is no longer the case, and at least the gaseous phase has to be treated as a compressible fluid. In this paper, we consider the coupling of a compressible flow region to an incompressible one based on a splitting of the pressure into a thermodynamic and a hydrodynamic part. The compressible Euler equations are then connected to the Mach number zero limit equations in the other region. These limit equations can be solved analytically in one space dimension that allows to couple them to the solution of a half‐Riemann problem on the compressible side with the help of velocity and pressure jump conditions across the interface. At the interface location, the flux terms for the compressible flow solver are provided by the coupling algorithms. The coupling is demonstrated in a one‐dimensional framework by use of a discontinuous Galerkin scheme for compressible two‐phase flow with a sharp interface tracking via a ghost‐fluid type method. The coupling schemes are applied to two generic test cases. The computational results are compared with those obtained with the fully compressible two‐phase flow solver, where the Mach number zero limit is approached by a weakly compressible fluid. For all cases, we obtain a very good agreement between the coupling approaches and the fully compressible solver. Copyright © 2014 John Wiley & Sons, Ltd.     

Solidification modelling with a control volume method on domains subjected to viscoplastic deformation

The rise in importance of semi-solid based products has created a need for accurate modelling approaches to coupled solidification and deformation. Current approaches to solidification modelling, using the finite element method (FEM), are principally founded on capacitance methods. Unfortunately they suffer from a major drawback in that energy is not correctly transported through elements, so providing a source of inaccuracy. This paper is concerned with the development and application of a control volume capacitance method (CVCM) to problems where viscoplastic deformation and solidification are combined. The approach adopted is founded on the theory that describes energy transfer through a control volume (CV) moving relative to the deforming mass. This essentially arbitrary Lagrangian–Eulerian (ALE) method facilitates the accurate treatment of discontinuities. The CV approach is tested against known analytical solutions and is shown to be accurate, stable and computationally competitive.     

A comparative study of transient capillary rise using direct numerical simulations

The rise of liquid in capillaries, or between two parallel plates as the 2D variant thereof, represents a challenging test case for two-phase flow solvers without a full analytic solution. Four different numerical approaches are compared for the rise of liquid, also providing reference data being of high relevance for capillarity-dominated wetting processes. The used methods are an Arbitrary Lagrangian–Eulerian method (OpenFOAM solver interTrackFoam), a geometric Volume of Fluid code (FS3D), an algebraic Volume of Fluid method (OpenFOAM solver interFoam), and a level-set based extended discontinuous Galerkin discretization (BoSSS).While the transient rise height shows excellent agreement between the different implementations, the velocity fields at the interface demonstrate a different level of local accuracy of the available approaches. Reducing the slip length reduces the overall dynamics of the system, thus yielding a qualitative change in the rise behavior – a behavior that is not covered by simplified ODE models. The obtained rise height results are vailable online: http://dx.doi.org/10.25534/tudatalib-173     

Coupling Boundary Elements to a Raytracing Procedure using the Singular Indirect Method

The coupling of Boundary Elements and a raytracing procedure is presented here. Such a hybrid method is best suited to the study of realistic outdoor sound propagation problems: The noise often acts in a domain where many objects like buildings or sound insulation walls scatter the sound. Thus, diffraction has to be taken into account. BEM is well suited. To study the effects of this noise on a sound receiver far away, raytracing may be preferable for such application, because refraction can be implemented more easily. Hence, a Boundary Element Analysis is performed in noisy nearfield regions, a raytracing procedure at a larger distance from the sound sources. First, the direct Boundary Element algorithm is applied to determine the sound pressure at interface points. Second, a singular indirect Boundary Element formulation is used to find intensities of point sources on the same interface which produce the previously determined sound pressure. Finally, these sound intensities are the input data for the raytracing procedure. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Penalty-combined approaches to the Ritz-Galerkin and finite element methods for singularity problems of elliptic equations

Penlty coupling techniques on an interface boundary, artificial or material, are first presented for combining the Ritz–Galerkin and finite element methods. An optimal convergence rate first is proved in the Sobolev norms. Moreover, a significant coupling strategy, L + 1 = O(|ln h|), between these two methods are derived for the Laplace equation with singularities, where L + 1 is the total number of particular solutions used in the Ritz–Galerkin method, and h is the maximal boundary length of quasiuniform elements used in the linear finite element method. Numreical experiments have been carried out for solving the benchmark model: Motz's problem. Both theoretical analysis and numreical experiments clearly display the importance of penalty-combined methods is solving elliptic equations with singularities.     

Further results of existence of positive solutions of elliptic Kirchhoff equation with general nonlinearity of source terms

The paper deals with some existence results for elliptic Kirchhoff equations with respect to general nonlinear source terms and changing sign data, where we have used three different methods: direct variational method, Galerkin approach, and subsolution–supersolution method.     

Reduced-dissipation remapping of velocity in staggered arbitrary Lagrangian-Eulerian methods

Remapping is an essential part of most Arbitrary Lagrangian-Eulerian (ALE) methods. In this paper, we focus on the part of the remapping algorithm that performs the interpolation of the fluid velocity field from the Lagrangian to the rezoned computational mesh in the context of a staggered discretization. Standard remapping algorithms generate a discrepancy between the remapped kinetic energy, and the kinetic energy that is obtained from the remapped nodal velocities which conserves momentum. In most ALE codes, this discrepancy is redistributed to the internal energy of adjacent computational cells which allows for the conservation of total energy. This approach can introduce oscillations in the internal energy field, which may not be acceptable. We analyze the approach introduced in Bailey (1984) [11] which is not supposed to introduce dissipation. On a simple example, we demonstrate a situation in which this approach fails. A modification of this approach is described, which eliminates (when it is possible) or reduces the energy discrepancy.     

FEM-BEM coupling with non-conforming interfaces

In order to treat wave propagation phenomena in coupled domains, a combined approach of Finite Element Methods (FEM) and Boundary Element Methods (BEM) is presented. The coupling is done within the framework of Tearing and Interconnecting methods (FETI/BETI), which are special non-overlapping domain decomposition methods. The coupling conditions are incorporated in a weak sense, which allows non-conforming interface discretization, i.e., the Mortar Method is used. A numerical example is given to verify the algorithm. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Strong stability of nonlinear semigroups with weak dissipation and non-compact resolvent–applications to structural acoustics

We consider asymptotic stability, in the strong topology, of a nonlinear coupled system of partial differential equations (PDEs) arising in structural–acoustic interactions. The coupling involves parabolic and hyperbolic dynamics with interaction on an interface–a manifold of lower dimension. The distinctive feature of the model is that the resolvent associated with the generator governing the evolution is not compact and the dissipation considered is ‘weak’. Thus, strong stability is not to be generally expected. In linear problems this difficulty is circumvented by the use of Taubrien theorems and spectral analysis [W. Arendt and C.J.K. Batty, Tauberian theorems and stability of one-parameter semi-groups, Trans. Amer. Math. Soc. 306(8) (1988), pp. 837–852, Y.I. Lyubich and V.Q. Phong, Asymptotic stability of linear differential equations ain Banach spaces, Studia Math., LXXXXVII, (1988), pp. 37–42, G.M. Sklyar, On the maximal asymptotica for linear equations in Banach spaces, 2009]. However these methods are not applicable to nonlinear dynamics.

In this article, we present an approach to strong stability that is applicable to nonlinear semigroups governed by multivalued generators with non-compact resolvents. The method relies on a suitable relaxation of Lasalle invariance principle [J.P. LaSalle, Stability theory and invariance principles, in Dynamical Systems, Vol. 1, L. Cerasir, J.K. Hale, J.P. LaSalle, eds., Academic Press, New York, 1976, pp. 211–222] which then requires appropriate unique continuation theorems along with a string of a-priori PDE estimates specific to parabolic–hyperbolic coupled systems.     

A more meaningful parameterization of the Lee–Carter model

A new Lee–Carter model parameterization is introduced with two advantages. First, the Lee–Carter parameters are normalized such that they have a direct and intuitive interpretation, comparable across populations. Second, the model is stated in terms of the “needed-exposure” (NE). The NE is the number required in order to get one expected death and is closely related to the “needed-to-treat” measure used to communicate risks and benefits of medical treatments. In the new parameterization, time parameters are directly interpretable as an overall across-age NE. Age parameters are interpretable as age-specific elasticities: percentage changes in the NE at a particular age in response to a percent change in the overall NE. A similar approach can be used to confer interpretability on parameters of other mortality models.     

Application of the Level-Set Model with Constraints in Image Segmentation

We propose and analyze a constrained level-set method for semi-automatic image segmentation. Our level-set model with constraints on the level-set function enables us to specify which parts of the image lie inside respectively outside the segmented objects. Such a-priori information can be expressed in terms of upper and lower constraints prescribed for the level-set function. Constraints have the same conceptual meaning as initial seeds of the popular graph-cuts based methods for image segmentation. A numerical approximation scheme is based on the complementary-finite volumes method combined with the Projected successive overrelaxation method adopted for solving constrained linear complementarity problems. The advantage of the constrained level-set method is demonstrated on several artificial images as well as on cardiac MRI data.