Efficient finite element formulation for geothermal heating systems. Part I: steady state

This paper presents the development of a computationally efficient finite element tool for the analysis of 3D steady state heat flow in geothermal heating systems. Emphasis is placed on the development of finite elements for vertical borehole heat exchangers and the surrounding soil layers. Three factors have contributed to the computational efficiency: the proposed mathematical model for the heat exchanger, the discretization of the spatial domain using the Petrov–Galerkin method and the sequential numerical algorithm for solving the resulting system of non‐linear equations. These have contributed in reducing significantly the required number of finite elements necessary for describing the involved systems. Details of the mathematical derivations and some numerical examples are presented. Copyright © 2005 John Wiley & Sons, Ltd.     

Higher‐order boundary element methods for transient diffusion problems. Part II: Singular flux formulation

A boundary element method (BEM) for transient heat diffusion phenomena presented in Part I is extended to problems involving instantaneous rise of temperature on a portion of the boundary. The new boundary element formulation involves the use of an infinite flux function in order to properly capture the singular response of the flux. It is shown that the conventional finite flux BEM formulation, as well as a commercial FEM code, results in a large first‐time‐step numerical error that cannot be reduced by mesh or time‐step refinement. The use of the singular flux formulation for BEM demonstrates an extremely high level of accuracy for the one‐dimensional case, and a significant improvement in the solutions within a two‐dimensional representation. The additional errors arising due to improper time interpolation of the temperature on the boundaries adjacent to the singular flux boundary are discussed. Copyright © 2002 John Wiley & Sons, Ltd.     

Energy–momentum consistent finite element discretization of dynamic finite viscoelasticity

This paper is concerned with energy–momentum consistent time discretizations of dynamic finite viscoelasticity. Energy consistency means that the total energy is conserved or dissipated by the fully discretized system in agreement with the laws of thermodynamics. The discretization is energy–momentum consistent if also momentum maps are conserved when group motions are superimposed to deformations. The performed approximation is based on a three‐field formulation, in which the deformation field, the velocity field and a strain‐like viscous internal variable field are treated as independent quantities. The new non‐linear viscous evolution equation satisfies a non‐negative viscous dissipation not only in the continuous case, but also in the fully discretized system. The initial boundary value problem is discretized by using finite elements in space and time. Thereby, the temporal approximation is performed prior to the spatial approximation in order to preserve the stress objectivity for finite rotation increments (incremental objectivity). Although the present approach makes possible to design schemes of arbitrary order, the focus is on finite elements relying on linear Lagrange polynomials for the sake of clearness. The discrete energy–momentum consistency is based on the collocation property and an enhanced second Piola–Kirchhoff stress tensor. The obtained coupled non‐linear algebraic equations are consistently linearized. The corresponding iterative solution procedure is associated with newly proposed convergence criteria, which take the discrete energy consistency into account. The iterative solution procedure is therefore not complicated by different scalings in the independent variables, since the motion of the element is taken into account for solving the viscous evolution equation. Representative numerical simulations with various boundary conditions show the superior stability of the new time‐integration algorithm in comparison with the ordinary midpoint rule. Both the quasi‐rigid deformations during a free flight, and large deformations arising in a dynamic tensile test are considered. Copyright © 2009 John Wiley & Sons, Ltd.     

Time‐dependent shape functions for modeling highly transient geothermal systems

This paper presents new time‐dependent finite element shape functions suitable for modeling high‐gradient transient conductive heat flow in geothermal systems. The shape functions are made adaptive by enhancing the approximation functions with time‐dependent variables, which may vary according to the transient process without adding extra degrees of freedom or applying mesh adaptation. Two different approaches are presented. First, an iterative method is proposed, in which an exponential approximation function, which is optimized continually during the transient process, is incorporated in the shape function. Second, an analytical method is suggested, in which an analytical solution of a simplified process is incorporated in the shape function, enabling an explicit update of the shape functions in each time step. A methodology for modeling the variation of temperature in one and two dimensions is introduced. The ability of the method to capture high‐gradient temperature profiles using relatively large elements is illustrated with numerical examples of cases in which equally large standard finite elements fail. Copyright © 2008 John Wiley & Sons, Ltd.     

A combined space–time extended finite element method

The Newmark method for the numerical integration of second order equations has been extensively used and studied along the past fifty years for structural dynamics and various fields of mechanical engineering. Easy implementation and nice properties of this method and its derivatives for linear problems are appreciated but the main drawback is the treatment of discontinuities. Zienkiewicz proposed an approach using finite element concept in time, which allows a new look at the Newmark method. The idea of this paper is to propose, thanks to this approach, the use of a time partition of the unity method denoted Time Extended Finite Element Method (TX‐FEM) for improved numerical simulations of time discontinuities. An enriched basis of shape functions in time is used to capture with a good accuracy the non‐polynomial part of the solution. This formulation allows a suitable form of the time‐stepping formulae to study stability and energy conservation. The case of an enrichment with the Heaviside function is developed and can be seen as an alternative approach to time discontinuous Galerkin method (T‐DGM), stability and accuracy properties of which can be derived from those of the TX‐FEM. Then Space and Time X‐FEM (STX‐FEM) are combined to obtain a unified space–time discretization. This combined STX‐FEM appears to be a suitable technique for space–time discontinuous problems like dynamic crack propagation or other applications involving moving discontinuities. Copyright © 2005 John Wiley & Sons, Ltd.     

A finite element approach to anisotropic damage of ductile materials in large deformations Part II: finite element formulation and applications

A finite element analysis model for material and geometrical non-linearities due to large plastic deformations of ductile materials is presented using the continuum damage mechanics approach. To overcome limitations of the conventional plastic analysis, a fourth-order tensor damage, defined in Part I of this paper to represent the stiffness degradation in the finite strain regime, is incorporated. General forms of an updated Lagrangian (U.L.) finite element procedure are formulated to solve the governing equations of the coupled elastic–plastic-damage analysis, and a computer program is developed for two-dimensional plane stress/strain problems. A numerical algorithm to treat the anisotropic damage is proposed in addition to the non-linear incremental solution algorithm of the U.L. formulation. Selected examples, compared with published results, show the validity of the presented finite element approach. Finally, the necking phenomenon of a plate with a hole is studied to explore plastic damage in large strain deformations. This revised version was published online in August 2006 with corrections to the Cover Date.     

A finite element formulation for thermoelastic damping analysis

We present a finite element formulation based on a weak form of the boundary value problem for fully coupled thermoelasticity. The thermoelastic damping is calculated from the irreversible flow of entropy due to the thermal fluxes that have originated from the volumetric strain variations. Within our weak formulation we define a dissipation function that can be integrated over an oscillation period to evaluate the thermoelastic damping. We show the physical meaning of this dissipation function in the framework of the well‐known Biot's variational principle of thermoelasticity. The coupled finite element equations are derived by considering harmonic small variations of displacement and temperature with respect to the thermodynamic equilibrium state. In the finite element formulation two elements are considered: the first is a new 8‐node thermoelastic element based on the Reissner–Mindlin plate theory, which can be used for modeling thin or moderately thick structures, while the second is a standard three‐dimensional 20‐node iso‐parametric thermoelastic element, which is suitable to model massive structures. For the 8‐node element the dissipation along the plate thickness has been taken into account by introducing a through‐the‐thickness dependence of the temperature shape function. With this assumption the unknowns and the computational effort are minimized. Comparisons with analytical results for thin beams are shown to illustrate the performances of those coupled‐field elements. Copyright © 2008 John Wiley & Sons, Ltd.     

Arbitrary discontinuities in space–time finite elements by level sets and X‐FEM

An enriched finite element method with arbitrary discontinuities in space–time is presented. The discontinuities are treated by the extended finite element method (X‐FEM), which uses a local partition of unity enrichment to introduce discontinuities along a moving hyper‐surface which is described by level sets. A space–time weak form for conservation laws is developed where the Rankine–Hugoniot jump conditions are natural conditions of the weak form. The method is illustrated in the solution of first order hyperbolic equations and applied to linear first order wave and non‐linear Burgers' equations. By capturing the discontinuity in time as well as space, results are improved over capturing the discontinuity in space alone and the method is remarkably accurate. Implications to standard semi‐discretization X‐FEM formulations are also discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

k‐version of finite element method in gas dynamics: higher‐order global differentiability numerical solutions

In this paper, we consider and examine alternate finite element computational strategies for time‐dependent Navier–Stokes equations describing high‐speed compressible flows with shocks in a viscous and conducting medium, with the ultimate objective of establishing the desired features of a general mathematical and computational framework for such initial value problems (IVP) in which: (a) the numerically computed solutions are in agreement with the physics of evolution described by the governing differential equations (GDEs) i.e. the IVP, (b) the solutions are admissible in the non‐discretized form of the GDEs in the pointwise sense (i.e. anywhere and everywhere) in the entire space–time domain, and hence in the integrated sense as well, (c) the numerical approximations progressively approach the same global differentiability in space and time as the theoretical solutions, (d) it is possible to time march the solutions (this is essential for efficiency as well as ensuring desired accuracy of the computed solution for the current increment of time, i.e. to minimize the error build up in the time marching process), (e) the computational process is unconditionally stable and non‐degenerate regardless of the choice of discretization, nature of approximations and their global differentiability and the dimensionless parameters influencing the physics of the process, (f) there are no issues of stability, CFL number limitations and (g) the mathematical and computational methodology is independent of the nature of the space–time differential operators. We consider one‐dimensional compressible flow in a viscous and conducting medium with shocks as model problems to illustrate various features of the general mathematical and computational framework used here and to demonstrate that the proposed framework is general and is applicable to all IVP. The Riemann shock tube with a single diaphragm serves as a model problem. The specific details presented in the paper discuss: (1) Choice of the form of the GDEs, i.e. strong form or weak form. (2) Various choices of variables. The paper establishes and considers density, velocity and temperature as variables of choice. (3) Details of the space–time least squares (LS) integral forms (meritorious over all others in all aspects) are presented and choice of approximation spaces are discussed. (4) In all numerical studies we consider a viscous and conducting medium with ideal gas law, however results are also presented for non‐conducting medium. Extension of this work to real gas models will be presented in a separate paper. It is worth noting that when the medium is viscous and conducting, the solutions of gas dynamics equations are analytic. (5) It is also significant to note that upwinding methods based on addition of artificial diffusion such as SUPG, SUPG/DC, SUPG/DC/LS and their many variations are neither needed nor used in this present work. (6) Numerical studies are aimed at resolving the localized details of the shock structure, i.e. shock relations, shock width, shock speed, etc. as well as the over all global behaviour of the solution in the entire space–time domain. (7) Numerical studies are presented for Riemann shock tube for high Mach number flows with special emphasis also on time accuracy of the evolution which is ensured by requiring that the approximations for each increment of time satisfy non‐discretized form of the GDEs in the pointwise sense, and hence in the integrated sense as well. (8) Comparisons are made with published results as well as theoretical solutions (when possible). It is established that space–time least squares processes are the only processes that yield variationally consistent space–time integral forms, and hence unconditionally non‐degenerate space–time computational processes, which when considered in higher‐order scalar product spaces provide the desired mathematical framework in which progressively higher‐order global differentiability solutions in space and time yield the same characteristics as the theoretical solutions of the IVP in all aspects. Copyright © 2006 John Wiley & Sons, Ltd.     

A rheological interface model and its space–time finite element formulation for fluid–structure interaction

This contribution discusses extended physical interface models for fluid–structure interaction problems and investigates their phenomenological effects on the behavior of coupled systems by numerical simulation. Besides the various types of friction at the fluid–structure interface the most interesting phenomena are related to effects due to additional interface stiffness and damping. The paper introduces extended models at the fluid–structure interface on the basis of rheological devices (Hooke, Newton, Kelvin, Maxwell, Zener). The interface is decomposed into a Lagrangian layer for the solid‐like part and an Eulerian layer for the fluid‐like part. The mechanical model for fluid–structure interaction is based on the equations of rigid body dynamics for the structural part and the incompressible Navier–Stokes equations for viscous flow. The resulting weighted residual form uses the interface velocity and interface tractions in both layers in addition to the field variables for fluid and structure. The weak formulation of the whole coupled system is discretized using space–time finite elements with a discontinuous Galerkin method for time‐integration leading to a monolithic algebraic system. The deforming fluid domain is taken into account by deformable space–time finite elements and a pseudo‐structure approach for mesh motion. The sensitivity of coupled systems to modification of the interface model and its parameters is investigated by numerical simulation of flow induced vibrations of a spring supported fluid‐immersed cylinder. It is shown that the presented rheological interface model allows to influence flow‐induced vibrations. Copyright © 2010 John Wiley & Sons, Ltd.     

The enriched space–time finite element method (EST) for simultaneous solution of fluid–structure interaction

The paper introduces a weighted residual‐based approach for the numerical investigation of the interaction of fluid flow and thin flexible structures. The presented method enables one to treat strongly coupled systems involving large structural motion and deformation of multiple‐flow‐immersed solid objects. The fluid flow is described by the incompressible Navier–Stokes equations. The current configuration of the thin structure of linear elastic material with non‐linear kinematics is mapped to the flow using the zero iso‐contour of an updated level set function. The formulation of fluid, structure and coupling conditions uniformly uses velocities as unknowns. The integration of the weak form is performed on a space–time finite element discretization of the domain. Interfacial constraints of the multi‐field problem are ensured by distributed Lagrange multipliers. The proposed formulation and discretization techniques lead to a monolithic algebraic system, well suited for strongly coupled fluid–structure systems. Embedding a thin structure into a flow results in non‐smooth fields for the fluid. Based on the concept of the extended finite element method, the space–time approximations of fluid pressure and velocity are properly enriched to capture weakly and strongly discontinuous solutions. This leads to the present enriched space–time (EST) method. Numerical examples of fluid–structure interaction show the eligibility of the developed numerical approach in order to describe the behavior of such coupled systems. The test cases demonstrate the application of the proposed technique to problems where mesh moving strategies often fail. Copyright © 2007 John Wiley & Sons, Ltd.     

An extended finite element formulation for contact in multi‐material arbitrary Lagrangian–Eulerian calculations

Multi‐material Eulerian and arbitrary Lagrangian–Eulerian methods were originally developed for solving hypervelocity impact problems, but they are attractive for solving a broad range of problems having large deformations, the evolution of new free surfaces, and chemical reactions. The contact, separation, and slip between two surfaces have traditionally been addressed by the mixture theory, however the accuracy of this approach is severely limited. To improve the accuracy, an extended finite element formulation is developed and example calculations are presented. As a side benefit, the mixture theory is eliminated from the multi‐material formulation, eliminating the issues associated with the equilibration time between adjacent materials. By design, the new formulation is relatively simple to implement in existing multi‐material codes, parallelizes without difficulty, and has a low memory burden. Copyright © 2006 John Wiley & Sons, Ltd.     

A contact algorithm for the Signorini problem using space–time finite elements

The most common approach in the finite‐element modelling of continuum systems over space and time is to employ the finite‐element discretization over the spatial domain to reduce the problem to a system of ordinary differential equations in time. The desired time integration scheme can then be used to step across the so‐called time slabs, mesh configurations in which every element shares the same degree of time refinement. These techniques may become inefficient when the nature of the initial boundary value problem is such that a high degree of time refinement is required only in specific spatial regions of the mesh. Ideally one would be able to increase the time refinement only in those necessary regions. We achieve this flexibility by employing space–time elements with independent interpolation functions in both space and time. Our method is used to examine the classic contact problem of Signorini and allows us to increase the time refinement only in the spatial region adjacent to the contact interface. We also develop an interface‐tracking algorithm that tracks the contact boundary through the space–time mesh and compare our results with those of Hertz contact theory. Copyright 2004 John Wiley & Sons, Ltd.     

陀螺系统时间有限元方法

将保辛的时间有限元方法应用于陀螺系统,扩展时间有限元方法的应用领域。同时导出陀螺系统时间有限元方法的形函数矩阵、时间单元刚度阵列式和非齐次外力的表达式。该方法既继承了有限元保辛的优良特性,又大大提高了数值计算精度,具有非常明显的优越性。算例给出本文方法、四阶Runge-Kutta方法与Newmark方法的比较结果,进一步表明本方法的优越性。     

A scaled boundary finite element formulation for dynamic elastoplastic analysis

This study presents the development of the scaled boundary finite element method (SBFEM) to simulate elastoplastic stress wave propagation problems subjected to transient dynamic loadings. Material nonlinearity is considered by first reformulating the SBFEM to obtain an explicit form of shape functions for polygons with an arbitrary number of sides. The material constitutive matrix and the residual stress fields are then determined as analytical polynomial functions in the scaled boundary coordinates through a local least squares fit to evaluate the elastoplastic stiffness matrix and the residual load vector semianalytically. The treatment of the inertial force within the solution of the nonlinear system of equations is also presented within the SBFEM framework. The nonlinear equation system is solved using the unconditionally stable Newmark time integration algorithm. The proposed formulation is validated using several benchmark numerical examples.     

地热能供热技术研究现状及展望

地热能因具有稳定、储量大、分布广泛等特点,在建筑供热领域得到广泛应用。本文从地热能供热技术分类出发,详细阐述了浅层地源热泵技术、水热型供热技术及中深层地埋管供热技术的基本概念、发展沿革及应用现状。通过对现有报道的地热能供热技术研究方向和相关成果进行总结,从运行机理和应用实践角度出发,对该领域技术发展做出展望。未来地热能供热技术领域研究方向主要包括大型浅层地埋管管群热平衡分析、水热型供热高效回灌技术研究及中深层地埋管管群换热性能评估等。     

Space–time approach to numerical analysis of a string with a moving mass

Inertial loading of strings, beams and plates by mass travelling with near‐critical velocity has been a long debate. Typically, a moving mass is replaced by an equivalent force or an oscillator (with ‘rigid’ spring) that is in permanent contact with the structure. Such an approach leads to iterative solutions or imposition of artificial constraints. In both cases, rigid constraints result in serious computational problems. A direct mass matrix modification method frequently implemented in the finite element approach gave reasonable results only in the range of relatively low velocities. In this paper we present the space–time approach to the problem. The interaction of the moving mass/supporting structure is described in a local coordinate system of the space–time finite element domain. The resulting characteristic matrices include inertia, Coriolis and centrifugal forces. A simple modification of matrices in the discrete equations of motion allows us to gain accurate analysis of a wide range of velocities, up to the velocity of the wave speed. Numerical examples prove the simplicity and efficiency of the method. The presented approach can be easily implemented in the classic finite element algorithms. Copyright © 2008 John Wiley & Sons, Ltd.     

A scaled boundary finite element formulation for poroelasticity

This paper develops the scaled boundary finite element formulation for applications in coupled field problems, in particular, to poroelasticity. The salient feature of this formulation is that it can be applied over arbitrary polygons and/or quadtree decomposition, which is widely employed to traverse between small and large scales. Moreover, the formulation can treat singularities of any order. Within this framework, 2 sets of semianalytical, scaled boundary shape functions are used to interpolate the displacement and the pore fluid pressure. These shape functions are obtained from the solution of vector and scalar Laplacian, respectively, which are then used to discretise the unknown field variables similar to that of the finite element method. The resulting system of equations are similar in form as that obtained using standard procedures such as the finite element method and, hence, solved using the standard procedures. The formulation is validated using several numerical benchmarks to demonstrate its accuracy and convergence properties.     

Efficient numerical strategies for spectral stochastic finite element models

The use of spectral stochastic finite element models results in large systems of equations requiring specialized solution strategies. This paper discusses three different numerical algorithms for solving these large systems of equations. It presents a trade‐off of these algorithms in terms of memory usage and computation time. It also shows that the structure of the spectral stochastic stiffness matrix can be exploited to accelerate the solution process, while keeping the memory usage to a minimum. Copyright © 2005 John Wiley & Sons, Ltd.