基于时域间断有限元方法的生物组织非傅立叶热行为数值分析

在热传导分析中,当热流与温度梯度存在时间延迟时,需采用非傅立叶热传导模型进行分析。生物组织具有较强的热松弛时间系数,承受激光、微波及烧烫等作用时,其呈现出较强的非傅立叶行为。本文对脉冲热源作用下生物组织的非傅立叶热传导进行研究,针对强脉冲引起的温度场在空间域的高梯度变化、波阵面的间断行为以及通用传统时域数值方法会带来虚假数值振荡的特点,提出采用所发展的时域间断Galerkin有限元法（DG-FEM ）进行求解计算。对多种脉冲热源作用下的非傅立叶热传导过程进行数值模拟,通过考量强脉冲作用下温度场分布和热致生物组织损伤行为的影响,表明了本文所发展的DGFEM 能够有效、准确地描述温度场空间分布和热传导过程以及非傅立叶行为下的生物热损伤更为明显,在生物组织热行为分析中应该受到重视。     

不连续温度场问题的间断Galerkin方法

针对不连续温度场问题建立了一种间断Galerkin有限元方法,该方法的主要特点是允许插值函数在单元边界上存在跳变.在建立有限元方程时,通过在单元边界上引入数值通量项和稳定性项来处理间断效应,并且数值通量可以直接由接触热阻的定义式导出.数值算例表明该方法可以很方便且准确地捕捉到结构内部由于接触热阻而引起的温度跳变,同时在局部高梯度温度场的模拟方面也比常规连续Galerkin有限元方法效率明显要高.该方法也为研究由接触热阻引起的温度场与应力场之间的耦合问题提供了一种新的数值模拟手段.     

间断Galerkin有限元和有限体积混合计算方法研究

通过局部坐标变换而建立的非正交单元间断Galerkin(DG)有限元计算方法计算精度高， 但计算量大、内存需求大；而非结构网格有限体积方法虽然准确计算热流的问题目 前还没有完全解决，但其具有计算速度快和内存需求小的优点. 该研究是将有 限元和有限体积方法的优点结合，发展有限元和有限体积的混合方法. 在物面 附近黏性占主导作用的区域内采用有限元方法进行计算，在远离物面的区域采用快速的有限 体积方法进行计算，在有限元和有限体积方法结合处要保证通量守恒. 通过算例说明有 限元和有限体积混合方法既能保证黏性区域的热流计算精度和流场结构的分辨率，又能 降低内存需求和提高计算效率，使有限元方法应用于复杂外形(实际工程问题)的计 算成为可能.     

水下声波传播过程的时域间断Galerkin有限元方法

本文构建了声压波动方程的改进时域间断Galerkin有限元方法.传统时域连续有限元方法在计算高梯度、强间断特征水中声波传播问题时往往会出现虚假数值振荡现象,这些数值振荡会影响正常波动的计算精度.为了解决这一问题,本文通过引入人工阻尼的方式构建了改进的时域间断Galerkin有限元方法,并针对具有高梯度、强间断特征的多障...     

适用于间断Galerkin方法的限制器研究

开发了一种适用于高精度间断Galerkin方法的斜率（多项式系数）限制器。与现有的斜率限制器不同,该限制器实施过程不考虑网格单元类型（三角形或四边形）,通过全微分方法构造新的多项式系数,因此,该限制器能够适用于各种类型网格——结构化网格、具有单一单元的非结构化网格和具有混合单元的非结构化网格。由于该限制器能够方便地应用于具有混合单元的非结构化网格,因此,本文使用的程序能够方便地求解具有复杂几何结构的流动问题。本文利用一些典型算例对其性能进行了验证,表明该限制器适用于不同类型的网格单元,能够在光滑解区保证高的精度,并能够在阊断区抑帛3非物理振荡。     

非傅里叶热弹性的时域间断迦辽金有限元方法

基于Lord-Shulman非傅里叶热弹性模型,提出了采用修正的时域间断迦辽金有限元方法(time discontinuousGalerkin finite element method, DGFEM)求解方法. DGFEM对温度场、位移场基本未知向量及其时间导数向量在时域中分别插值;在最终的求解公式中,引入了人工阻尼. 数值结果显示所发展的DGFEM 较好地捕捉了波的间断并消除了热冲击作用下虚假的数值振荡,能够良好地模拟热弹性问题并具有较高的精度.     

改进时域间断Galerkin有限元方法在弹塑性波传播数值模拟中的应用

对于高频、强脉动荷载作用下的结构动力学波传播分析,对比于传统的时域算法,时域间断Galerkin方法能捕捉到波阵面的间断,有效地避免了由于间断引起的数值振荡.但时域间断方法却带来了波前面的虚假数值振荡.论文针对上述波前数值振荡的现象进行研究,通过引入人工阻尼的方法对时域间断Galerkin有限元方法进行进一步改进.数值结果表明,所发展的方法能够有效的滤掉强动荷载产生的波前数值振荡现象,同时降低了时域间断Galerkin方法的网格依赖性.     

一类指数型间断函数的嵌入非连续等参有限元法

在嵌入非连续有限元的基本思想下,提出一类附加位移形函数———指数型间断函数,来模拟由于非连续结构,如裂纹和节理,所导致的位移不连续规律,该附加函数是以到间断处的垂直距离为自变量,且随距离的增大而呈指数衰减的函数.指数型间断函数具有在数学上的便于积分和求导的优点,且比阶梯间断函数更能反映实际破裂后的变形情况.本文用弱解形式推导了嵌入非连续有限元格式,编制了二维4节点和三维8节点的嵌入非连续等参有限元程序,并分别给出了算例.算例表明在模拟裂纹追踪时,指数型间断函数的嵌入非连续等参有限元法可行且有效.     

弹塑性体中波传播问题的间断Galerkin有限元法

对结构动力学和波传播问题提出了一个时域间断的Galerkin有限元法.其主要特点是对问题的半离散场方程的节点基本未知向量及其时间导数向量在时间域中分别采用三次多项式和线性(P3-P1)插值,节点基本未知(位移)向量在离散的时间段之间将自动保证连续,而仅仅是它的时间导数(速度)向量存在间断.在非线性条件下,与现有的间断Galerkin有限元法相比,明显地节省了计算工作量.对所提出的间断Galerkin有限元法发展了弹塑性非线性问题的隐式和显式算法.数值计算结果表明了所提出方法的有效性,以及相对基于连续Galerkin有限元法的Newmark算法的计算结果的优越性.     

基于非傅立叶热传导半无限大体热冲击力学分析

在微观尺度下,陶瓷的热传导机理与宏观尺度下的热传导机理有较大的差异,由此产生的热应力与宏观尺度下的不同,本文针对热冲击条件下的半无限大体,基于非傅立叶热传导模型分析了温度场、应力场和单边裂纹的应力强度因子,并与基于傅立叶热传导模型分析的结果进行了比较.研究表明在半无限大体表面附近或裂纹较短时,基于非傅立叶热传导模型得到的应力最大值或应力强度因子最大值比基于傅立叶热传导模型得到的结果大,而在半无限大体内部或裂纹较长时,结果相反.     

Time discontinuous Galerkin finite element method for dynamic analyses in saturated poro-elasto-plastic medium

A time-discontinuous Galerkin finite element method for dynamic analyses in saturated poro-elasto-plastic medium is proposed. As compared with the existing discontinuous Galerkin finite element methods, the distinct feature of the proposed method is that the continuity of the displacement vector at each discrete time instant is automatically ensured, whereas the discontinuity of the velocity vector at the discrete time levels still remains. The computational cost is then obviously reduced, particularly, for material non-linear problems. Both the implicit and explicit algorithms to solve the derived formulations for material non-linear problems are developed. Numerical results show a good performance of the present method in eliminating spurious numerical oscillations and providing with much more accurate solutions over the traditional Galerkin finite element method using the Newmark algorithm in the time domain. The project supported by the National Natural Science Foundation of China (19832010, 50278012, 10272027) and the National Key Basic Research and Development Program (973 Program, 2002CB412709)     

A spectral‐element discontinuous Galerkin thermal lattice Boltzmann method for conjugate heat transfer applications

We present a spectral‐element discontinuous Galerkin thermal lattice Boltzmann method for fluid–solid conjugate heat transfer applications. Using the discrete Boltzmann equation, we propose a numerical scheme for conjugate heat transfer applications on unstructured, non‐uniform grids. We employ a double‐distribution thermal lattice Boltzmann model to resolve flows with variable Prandtl (Pr) number. Based upon its finite element heritage, the spectral‐element discontinuous Galerkin discretization provides an effective means to model and investigate thermal transport in applications with complex geometries. Our solutions are represented by the tensor product basis of the one‐dimensional Legendre–Lagrange interpolation polynomials. A high‐order discretization is employed on body‐conforming hexahedral elements with Gauss–Lobatto–Legendre quadrature nodes. Thermal and hydrodynamic bounce‐back boundary conditions are imposed via the numerical flux formulation that arises because of the discontinuous Galerkin approach. As a result, our scheme does not require tedious extrapolation at the boundaries, which may cause loss of mass conservation. We compare solutions of the proposed scheme with an analytical solution for a solid–solid conjugate heat transfer problem in a 2D annulus and illustrate the capture of temperature continuities across interfaces for conductivity ratio γ > 1. We also investigate the effect of Reynolds (Re) and Grashof (Gr) number on the conjugate heat transfer between a heat‐generating solid and a surrounding fluid. Steady‐state results are presented for Re = 5?40 and Gr = 10⁵?10⁶. In each case, we discuss the effect of Re and Gr on the heat flux (i.e. Nusselt number Nu) at the fluid–solid interface. Our results are validated against previous studies that employ finite‐difference and continuous spectral‐element methods to solve the Navier–Stokes equations. Copyright © 2016 John Wiley & Sons, Ltd.     

A moving discontinuous Galerkin finite element method for flows with interfaces

A moving discontinuous Galerkin finite element method with interface condition enforcement is formulated for flows with discontinuous interfaces. The underlying weak formulation enforces the interface condition separately from the conservation law, so that the residual only vanishes upon satisfaction of both. In this formulation, the discrete grid geometry is treated as a variable, so that, in contrast to the standard discontinuous Galerkin method, this method has both the means to detect interfaces, via interface condition enforcement, and to satisfy, via grid movement, the conservation law and its associated interface condition. The method therefore directly fits interfaces, including shocks, preserving a high-order representation up to the interface without requiring shock capturing or an upwind numerical flux to achieve stability. It can be generalized to flows with a priori unknown interfaces with nontrivial topology and curved interface geometry as well as to an arbitrary number of spatial dimensions. Unsteady flows are represented in a manner similar to steady flows using a space-time formulation. In addition to computing flows with interfaces, the method can represent point singularities in a flow field by degenerating cuboid elements. In general, the method works in conjunction with standard local grid operations, including edge collapse, to ensure that degenerate cells are removed. Test cases are presented for up to three-dimensional flows that provide an initial assessment of the stability and accuracy of the method.     

Coupling of finite element method and discontinuous Galerkin method to simulate viscoelastic flows

In this paper, a numerical method, which is about the coupling of continuous and discontinuous Galerkin method based on the splitting scheme, is presented for the calculation of viscoelastic flows of the Oldroyd‐B fluid. The momentum equation is discretized in time by using the Adams‐Bashforth second‐order algorithm, and then decoupled via the splitting approach. Considering the Oldroyd‐B constitutive equation, the second‐order Runge‐Kutta approach is selected to complete the temporal discretization. As for the spatial discretizations, the fundamental purpose is to make the best of finite element method (FEM) and discontinuous Galerkin (DG) method to handle different types of equations. Specifically speaking, for the subequations, FEM is chosen to treat the Poisson and Helmholtz equations, and DG is employed to deal with the nonlinear convective term. In addition, because of the hyperbolic nature, DG is also utilized to discretize the Oldroyd‐B constitutive equation spatially. This coupled method avoids resorting to extra stabilization technique occurred in standard FEM framework even for moderately high values of Weissenberg number and also reduces the complexity compared with unified DG scheme. The Oldroyd‐B model is applied to investigate several typical and challenging benchmarks, such as the 4:1 planar contraction flow and the lid‐driven cavity flow, with a wide range of Weissenberg number to illustrate the feasibility, robustness, and validity of our coupled method.     

A high‐order accurate discontinuous Galerkin finite element method for laminar low Mach number flows

In this paper we present a discontinuous Galerkin (DG) method designed to improve the accuracy and efficiency of laminar flow simulations at low Mach numbers using an implicit scheme. The algorithm is based on the flux preconditioning approach, which modifies only the dissipative terms of the numerical flux. This formulation is quite simple to implement in existing implicit DG codes, it overcomes the time‐stepping restrictions of explicit multistage algorithms, is consistent in time and thus applicable to unsteady flows. The performance of the method is demonstrated by solving the flow around a NACA0012 airfoil and on a flat plate, at different low Mach numbers using various degrees of polynomial approximations. Computations with and without flux preconditioning are performed on different grid topologies to analyze the influence of the spatial discretization on the accuracy of the DG solutions at low Mach numbers. The time accurate solution of unsteady flow is also demonstrated by solving the vortex shedding behind a circular cylinder at the Reynolds number of 100. Copyright © 2012 John Wiley & Sons, Ltd.     

Discontinuous Galerkin spectral element lattice Boltzmann method on triangular element

Discontinuous Galerkin spectral element method is used to solve the lattice Boltzmann equation (LBE) in the discrete velocity space. The triangular elements are adopted because of their flexibility to deal with complex geometries. The flow past a circular cylinder is simulated by the proposed scheme. The results are consistent with those obtained from the previous numerical methods and experiments. Copyright © 2003 John Wiley & Sons, Ltd.     

NEW ALGORITHM OF COUPLING ELEMENT-FREE GALERKIN WITH FINITE ELEMENT METHOD

IntroductionMeshless methods, as a special numerical method, originated from1970s. Since thediffuse element method was proposed by Nayroleset al.[1]in1992, the meshless methodshave received wide attentions in the mechanics area, and have shown obvious adv…     

非均质材料热传导问题的扩展有限元法

针对非均质材料,提出了以导热系数为基本参数的热传导扩展有限元法。划分网格时不需要考虑材料界面的存在,因此网格的形成可以大大地简化,且可以获得高质量的网格。不含材料界面的单元,其温度场函数将退化为常规有限元的函数。含材料界面的单元,采用基于水平集的加强函数加强常规温度的近似,加强函数用于模拟界面。数值算例结果体现了该方法...     

热冲击作用下层合皮肤组织热传导过程数值模拟

在急剧温度变化等强间断温度冲击作用下的生物层合组织非傅里叶热传导分析中,经典时域连续有限元方法(如Newmark等方法)会在波阵面以后的和层合组织界面附近的区域表现出强烈的数值振荡。这类数值振荡会影响问题求解精度,并带来较大不确定性。针对这类现象,本文发展了改进时域间断Galerkin有限元方法,进一步开展了相关问题的数值模拟。其控制方程的基本未知数(温度)及其时间导数在指定时间间隔内假设存在间断且独立插值。在有限元离散列式中引入比例刚度阵人工阻尼,以成功消除波前位置的虚假数值振荡行为。通过算例对比分析,相比Newmark方法和传统间断Galerkin方法,所提出的改进时域间断Galerkin有限元方法较好消除了波前、波后以及组织界面处的数值振荡,有效捕捉了波阵面的间断行为,提高了计算的精度。