Making the semi-implicit space-time conservation-element solution-element method of Jerez et al. more explicit

In the beginning of the nineties a NASA-group around S. C. Chang started to work on a new method for unsteady flow computation with seemingly good results. Thereby elements are space-time domains. Within solution-elements state and flux variables are linearized satisfying the underlying differential equation, within conservation-elements space-time flux is conserved. Proceeding this way for one-dimensional pipe flow Jerez et al. include source terms with all cross-section dependencies making the method semi-implicit. We show that by taking simple measures as - accounting for cross-sections in state and flux variables of mass and energy or - subtracting spatially integrated source terms from the flux the method may become more or completely explicit, particularly helpful when chemical species transport is involved. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A family of ELLAM schemes for advection-diffusion-reaction equations and their convergence analyses

A family of ELLAM (Eulerian–Lagrangian localized adjoint method) schemes is developed and analyzed for linear advection-diffusion-reaction transport partial differential equations with any combination of inflow and outflow Dirichlet, Neumann, or flux boundary conditions. The formulation uses space-time finite elements, with edges oriented along Lagrangian flow paths, in a time–stepping procedure, where space-time test functions are chosen to satisfy a local adjoint condition. This allows Eulerian–Lagrangian concepts to be applied in a systematic mass-conservative manner, yielding numerical schemes defined at each discrete time level. Optimal-order error estimates and superconvergence results are derived. Numerical experiments are performed to verify the theoretical estimates. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 739–780, 1998     

Finite element methods for strongly-coupled systems of fluid-structure interaction with application to granular flow in silos

A monolithic approach to fluid-structure interactions based on the space-time finite element method is presented to investigate stress states in silos filled with granular material during discharge. The thin-walled silo-shell is discretized by continuum based, mixed-hybrid finite elements, whereas the flowing granular material is described by an enhanced viscoplastic non-Newtonian fluid model. To adapt the mesh nodes of the fluid domain to the structural deformations, a mesh-moving scheme using a pseudo-solid is applied. The level-set-method involving XFEM is used, including a 4D split algorithm for the space-time finite elements, in order to describe free surfaces. The method is applied to 3D silo discharges. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Adaptive higher-order space-time discontinuous Galerkin method for the computer simulation of variably-saturated porous media flows

This paper is concerned with the numerical simulation of time-dependent variably-saturated Darcian flow problems described by the Richards equation. We present the adaptive higher-order space-time discontinuous Galerkin (hp-STDG) method which optimizes accuracy and efficiency by balancing the errors that arise from the space and time discretizations and from the resulting nonlinear algebraic system. Convergence problems related to the transition between unsaturated flow and saturated flow are eliminated by regularizing the constitutive formulas. We also present an hp-anisotropic mesh adaptation technique capable of generating unstructured triangular elements with optimal sizes, shapes, and polynomial approximation degrees. Several numerical experiments are presented to demonstrate the accuracy, efficiency, and robustness of the numerical method presented here.     

The Surabale of Space-Time and Its Algebra Signature Split

A “surable” is a category given by a special manifold of geometric algebra frames. It is a bale brought on by a surjective map the equivalence classes of which can constitute base elements of the associative algebra. It is also a stranded braid of idempotents based on a sheaf of base unipotents. The stranded braid of idempotents which are thought to represent particles or fields consists of fibers strictly separated by mutual annihilation throughout the bundle. The surabale can be defined on the Clifford algebra of space-time. Then it constitutes a bundle of frames which – though covering all dimensions of the geometric algebra – turn out as isomorphic to the ground space generating the algebra. Because of this, the mass shell and the Dirac-Hestenes equation can be defined on the whole surabale. As a result the equation is preserved when acted on by the symmetries of the transformation group of the standard model. The K?hler-equation simply turns into a Dirac-Hestenes equation on the inhomogenous surabale, yet with the same simple differential 1-forms of the linear equation. This shows very beautifully that the equation of motion as well as the invariance of the surabale under the standard model symmetry can be formulated base free. The Clifford bases – instead of Gra?mann – just brings in the riches of representation, that is, the emergence of the standard model. But its grading, in a way, is an illusion. Studying the dimension of the space-time-split in quadratic Clifford algebras, it turns out that the dimension of the positive space-like component reproduces their period-8 properties. Considering as an example the Minkowski space-time in the Lorentz metric rather than in (+  −   −   − ) we can see that physicists found the electroweak symmetries in the negative part of the geometry, here denoted as , but did not realize the strong force symmetries in the positive part since those depend on the graded motion in the graded subspace . It is comparatively difficult to find the generators of the group capable to represent the classic SU(3) with its root space A₂. Though the approach put forward gives satisfying answers to some classical problems of relativistic quantum mechanics, it does not solve the most important riddle which has been variously pointed out by Professor Oziewicz, namely, mechanics is not governed by the Lorentz- or Poincaré-group. The simplest argument to be held against it, is that the Lorentz/Poincaré group by definition is the symmetry group of the metric tensor in the empty space-time without bodies and radiation. How can such a Bewegungsgruppe of the empty space-time be related to the physics and mechanics of material bodies? [1], [2] May be this first argument is not convincing enough. But Oziewicz has listed a considerable number of arguments concerning the whole observation process against the present day unquestioned but incorrect application of the full Lorentz group. To clarify this will still need some more fundamental efforts which do not concern the main subject of this paper.     

The generalized relations among the code elements for Fibonacci coding theory

We have considered a class of square Fibonacci matrix of order (p + 1) whose elements are based on the Fibonacci p numbers with determinant equal to +1 or −1. There is a relation between Fibonacci numbers with initial terms which is known as cassini formula. Fibonacci series and the golden mean plays a very important role in the construction of a relatively new space–time theory, which is known as E-infinity theory. An original Fibonacci coding/decoding method follows from the Fibonacci matrices. There already exists a relation between the code matrix elements for the case p = 1 [Stakhov AP. Fibonacci matrices, a generalization of the cassini formula and a new coding theory. Chaos, Solitons and Fractals 2006;30:56–66.]. In this paper, we have established generalized relations among the code matrix elements for all values of p. For p = 2, the correct ability of the method is 99.80%. In general, correct ability of the method increases as p increases.     

On vibrations of an inhomogeneous anisotropic plate

An equation of motion of a flat inhomogeheous anisotropic plate is considered. Formal asymptotic solutions are constructed by applying the space-time ray method. An equation describing the flow of energy is obtained in the form of a continuity equation. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 73–82. Translated by A. S. Golubeva.     

Determining the limits of geometrical tortuosity from seepage flow calculations in porous media.

Recent investigations have found a distinct correlation of effective properties of porous media to sigmoidal functions, where one axis is the Reynolds number Re and the other is the effective property dependent of Re, Θ_i = S_i(Re). One of these properties is tortuosity. At very low Re (seepage flow), there is a characteristic value of tortuosity, and it is the upper horizontal asymptote of the sigmoidal function. With higher values of Re (transient flow) the tortuosity value decreases, until a lower asymptote is reached (turbulent flow). Estimations of this parameter have been limited to the low Reynolds regime in the study of porous media. The current state of the art presents different numerical measurements of tortuosity, such as skeletization, centroid binding, and arc length of streamlines. These are solutions for the low Re regime. So far, for high Re, only the arc length of stream lines has been used to calculate tortuosity. The present approach involves the simulation of fluid flow in large domains and high Re, which requires numerous resources, and often presents convergence problems. In response to this, we propose a geometrical method to estimate the limit of tortuosity of porous media at Re → ∞, from the streamlines calculated at low Re limit. We test our method by calculating the tortuosity limits in a fibrous porous media, and comparing the estimated values with numerical benchmark results. Ongoing work includes the geometric estimation of different intrinsic properties of porous media. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

奇异摄动问题最优阶一致收敛的间断有限元分析

采用非对称内罚间断有限元方法(以下简称NIPG方法)求解一维对流扩散型奇异摄动问题.理论上证明了采用拉格朗日线性元的NIPG方法在Bakhvalov-Shishkin网格上具有最优阶的一致收敛性,即在能量范数度量下其误差估计为O(N~(-1)),其中N为网格剖分中单元个数.数值算例验证了理论分析的正确性.     

空间-时间分数阶对流扩散方程的数值解法

本文考虑一个空间-时间分数阶对流扩散方程.这个方程是将一般的对流扩散方程中的时间一阶导数用α(0＜α＜1)阶导数代替,空间二阶导数用β(1＜β＜2)阶导数代替.本文提出了一个隐式差分格式,验证了这个格式是无条件稳定的,并证明了它的收敛性,其收敛阶为O(ι h).最后给出了数值例子.     

Strongly-coupled modelling and analysis of energy harvesting devices

A monolithic approach is proposed that provides simultaneous modelling and analysis of the harvester, which involves surface-coupled fluid-structure interaction, volume-coupled electro- mechanics and a controlling energy harvesting circuit for applications in energy harvesting. A space-time finite element approximation is used for numerical solution of the weighted residual form of the governing equations of the flow-driven piezoelectric energy harvesting device. This method enables time-domain investigation of different types of structures (plate, shells) subject to exterior/interior flow with varying cross sections, material compositions, and attached electrical circuits with respect to the electrical power output generated. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

含开边界二维Stokes问题的Galerkin边界元解法

本文推导了含有开边界的二维有限域上Stokes问题的边界积分方程, 得出基于单层位势的第一类间接边界积分方程.对与之等价的边界变分方程用Galerkin边界元求解以得出单层位势的向量密度. 对于含有开边界端点的边界单元,采用特别的插值函数, 以模拟其固有的奇异性.论文用若干数值算例模拟了含有开边界的有限区域上不可压缩粘性流体的绕流.       

Advanced FEM for free surface flow with application to landslides

An advanced space-time finite element method is presented to investigate movements of landslides and their interaction with flexible structures. The mechanics of liquefied soil is described by Navier-Stokes-equations for visco-plastic non-newtonian fluid. Likewise the fluid the kinematics of the structure is described by velocities, taking large rotations into account. This leads to a monolithic fluid-structure interaction approach considering the multi-field problem as a whole. The discretized model equations are assembled in a single set of algebraic equations, which are solved by applying Newton-Raphson scheme. Free surface motion of landslide is described by the level-set method. To reduce computational effort the fragmented finite element method is used, where only active finite elements are evaluated. A pde-based extrapolation of the velocity-field is applied to ensure an accurate transport of distance function, which defines the profile in space and time of the free surfaces. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A space-time finite element method for fractional wave problems

Numerical Algorithms - In this paper, we analyze a space-time finite element method for fractional wave problems involving the time fractional derivative of order γ (1 < γ...     

Free convection heat and mass transfer in a power law fluid past an inclined surface with thermophoresis

The problem of heat and mass transfer in a power law, two-dimensional, laminar, boundary layer flow of a viscous incompressible fluid over an inclined plate with heat generation and thermophoresis is investigated by the characteristic function method. The governing non-linear partial differential equations describing the flow and heat transfer problem are transformed into a set of coupled non-linear ordinary differential equation which was solved using Runge–Kutta shooting method. Exact solutions for the dimensionless temperature and concentration profiles, are presented graphically for different physical parameters and for the different power law exponents 0 < n < 0.5 and for n > 0.5.     

序贯可加图的结构

给定一个(p,q)图G和一个正整数k，G的一个k序贯可加性编码是不同的数k,k+1,…,k+p+q-1到G的p+q个元素的一种分配，使得G的每一边e=uv得到分配给顶点u和v的数值之和。若图的元素容许有这样的一种分配，则称该图是一个k序贯可加图。该文将给出序贯可加图的若干结构性质，并构造一个k序贯可加图的无限簇。      

The non-homogeneous flow of a thixotropic fluid around a sphere

The non-homogeneous flow of a thixotropic fluid around a settling sphere is simulated. A four-parameter Moore model is used for a generic thixotropic fluid and discontinuous Galerkin method is employed to solve the structure-kinetics equation coupled with the conservation equations of mass and momentum. Depending on the normalized falling velocity U*, which compares the time scale of structure formation and destruction, flow solutions are divided into three different regimes, which are attributed to an interplay of three competing factors: Brownian structure recovery, shear-induced structure breakdown, and the convection of microstructures. At small U*( ≪ 1), where the Brownian structure recovery is predominant, the thixotropic effect is negligible and flow solutions are not too dissimilar to that of a Newtonian fluid. As U* increases, a remarkable structural gradient is observed and the structure profile around the settling sphere is determined by the balance of all three competing factors. For large enough U*( ≫ 1), where the Brownian structure recovery becomes negligible, the balance between shear-induced structure breakdown and the convection plays a decisive role in determining flow profile. To quantify the interplay of three factors, the drag coefficient Cs of the sphere is investigated for ranges of U*. With this framework, the effect of the destruction parameter, the confinement ratio, and a possible nonlinearity in the model-form on the non-homogeneous flow of a thixotropy fluid have been addressed.     

Nonstationary love waves of theSH type in an anisotropic elastic medium. A kinematic approach. II

In this paper, the study of the high-frequency Love waves (simular to the well-known transverse waves of theSH type) near the surface of an anisotropic elastic body is continued. The formulation of the boundary-value problem, independent of a specific form of the elasticity tensor, provides the possibility of developing a kinematic approach, which is essential for constructing the asymptotics of these high-frequency waves. To this end, an algorithm is proposed that allows one to relate the transversality of the polarization of surface waves to the directions of their propagation on the surface and to obtain the conditions necessary for the origination of such waves. The algorithm suggested makes it possible to indicate those types of symmetry of media (special cases of anisotropy) for which the directions obtained correspond to the field of rays of Love waves. In these cases, the space-time ray method provides a mathematical tool for constructing uniform asymptotics of the surface waves in question. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol 218, 1994, pp. 206–219. This work was supported by the Russian Foundation of Fundamental Research (Grant 93-011-16148). Translated by Z. A. Yanson.     

A numerical study of the dynamics of elastic articulated systems in a fluid

We propose a method and an algorithm for computing the dynamics of elastic structures of articulated form in a fluid flow taking account of the weakening in certain structural elements. In describing the motion we use two sets of radius-vectors, which are approximated in the computations by parametric local splines of first degree. The possibilities of the proposed method are illustrated using the example of the study of the dynamics of transition processes in an articulated anchor-buoy structure, which arise when there is an abrupt change in the direction of the fluid flow velocity. We determine the kinematic and force characteristics of the structure under various changes in the direction of the flow velocity. We determine the structural elements in which the weakening occurs. Three figures. Bibliography: 7 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 128–134.     

On Error Estimate of Space-Time Finite Element Methods for Parabolic Equations in a Time-Dependent Domain

We consider linear parabolic equations in a space-time domain with curved boundaries and nonhomogeneous Dirichlet boundary conditions and discuss their approximations with isoparametric space-time finite elements. A general error estimate is proved and applied to some elements of practical interest.