弹性材料及超弹性材料中波传播问题的间断Galerkin有限元方法
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摘要
本文主要针对材料学中三种材料：棒状非线性弹性复合材料、细柱状非凸弹性材料以及杆状可压缩超弹性材料中波在传播过程中追赶碰撞问题以及波的相变问题进行研究。利用(局部)间断Galerkin方法及自适应Runge-Kutta间断Galerkin (RKDG)方法进行数值模拟。本文内容大致分成四个部分。
第一部分针对棒状非线性弹性复合材料,其模型方程可以看成是守恒律方程组,但是由于不同材料的(两层)复合,模型方程中流通量项出现间断,且在模型方程的右端源项出现δ函数,这些都给数值格式的设计增加了困难。由于流通量项的不连续性,因此在处理数值流通量的时候需要注意。另外由于物理界面的存在,波发生反射和透射,因此方程的解会包含多个激波,从而也增加了问题的难度。我们首先从物理的角度,引入耗散率和物理容许解的概念,证明间断Galerkin (DG)格式的能量稳定性和DG解是物理容许解。然后借助于耗散率,我们提出一种dissipation-rate reserving DG格式,并通过数值算例比较说明该格式的优点。最后我们利用dissipation-rate reserving DG格式数值模拟了复合材料中波在传播过程中发生碰撞的细节,包括拉伸激波、压缩激波和稀疏波之间的碰撞,并给出了详细的碰撞时间和位置；通过选取四组不同参数比较碰撞之后拉伸波的幅值减小率来说明参数的影响,结果也验证了材料的非线性性能够使得材料中具有破坏性的拉伸波降低幅值达400%。这也表明这样的非线性弹性材料具有很好的抗冲击能力。最后我们考虑应力参数ε以及右边界对波的影响,还有四层复合材料中材料排列方式对材料保护的影响。
第二部分针对细柱状非凸弹性材料,由于应力-应变关系的非凸性,将会导致解的不适定性,因此在材料中考虑柱体径向变形的影响得到新的模型方程,正是这种影响使得新的模型方程中包含高阶时间导数项,从而增加求解的难度。为了便于使用局部间断Galerkin (LDG)方法进行求解,我们引入新的时间辅助变量从而将模型方程转化为只具有一阶的时间导数的方程组,针对这-方程组设计了LDG方法。其次证明了LDG格式的稳定性,并通过数值算例给出LDG格式的精度：在L1模和L∞模意义下应变γ具有k+1阶精度。最后利用LDG格式进行了一系列数值模拟,由于色散和材料非凸性的影响,我们观察到了一些有趣的波动模式(wave pattern),例如,transformation front与类似于孤立子的波组成的波动模式和稀疏波与类似于孤立子的波组成的波动模式等等：另外我们给出了稀疏波与transformation fronts碰撞的详细过程。
第三部分针对杆状可压缩超弹性材料,其模型方程(可压缩超弹性杆波动方程)与Camassa-Holm方程形式很一致,但是在非线性项多一个常数系数γ。正是由于常数系数γ的存在,使得模型方程不再具有好的性质(例如可积性)。模型方程所具有的多种类型波(例如紧孤立波,孤立子波波)是我们重点考察对象,我们利用LDG格式数值模拟这些类型波之间的相互碰撞,发现一些有趣的现象：紧孤立波、孤立子波在Gauss扰动下仍然是稳定的；两个相向的紧孤立波之间碰撞后以及两个右行孤立子波之间碰撞后均有类似于cuspon波形出现；两个紧孤立波碰撞后会出现两个孤立子波；三个紧孤立波碰撞后会出现三个大小不同的孤立子波；一个右行紧孤立波与一个孤立子波碰撞后会出现两个大小不同的孤立子波。
第四部分同样针对棒状非线性弹性复合材料波的传播问题,从第一部分的数值计算结果可以看出应变γ包含多个激波,因此采用较大数目的网格进行模拟来抓住激波的信息,但是网格数目增大同时也大大增加了计算量。因为DG方法具有易于实现自适应的优点,我们采用自适应RKDG方法并结合KXRCF指示子(indicator)和TVB指示子这两种不同的指示子进行模拟,对是否需要TVB重构、参数M、有无耗散DG格式以及Pk元(所有次数不超过k的多项式)等多方面的影响进行考虑。总体而言,TVB指示子的计算结果要比KXRCF指示子好,因为存在的振荡少些。但是激波处出现少许光滑化的现象,因此还需通过选取合适的M值以使得结果更为精确。
In this thesis, we mainly focus on three types of materials:nonlinearly elastic composite bar, non-convex elastic slender cylinder and compressible hyper-elastic rod, and study the wave catching-up and collision phenomena and the phase transition in wave propagation in these materials. We use (local) discontinuous Galerkin method and adaptive Runge-Kutta discontinuous Galerkin method for numerical simulation.
There are four parts in this thesis. In the first part, we consider nonlinearly e-lastic composite bar, in which the model equations is conservation laws. The model equations contain a discontinuous flux and a source term with a δ-function due to the materials'composition. These set difficulties for the design of numerical schemes. Some attention should be paid to deal with the numerical fluxes for the discontinuity of the flux. In addition, the waves can reflect and transmit at the physical interface, and the solutions contain many shock waves, which also increase the difficulty for the problem. From the point of physics, we first introduce the concept of dissipation rate and physically admissible solution. Then, by virtue of the dissipation rate, we develop a dissipation-rate reserving discontinuous Galerkin method and give numerical simula-tions to demonstrate the advantage of the scheme. In the end, we use the dissipation-rate reserving discontinuous Galerkin scheme to simulate the details of the interactions of waves at the propagation process, which include the interactions between tensile shock wave, compressive shock and rarefaction wave. We give the detailed times and po-sitions for the interactions. We also examine how material parameters influence the reduction ratio by four different group of parameters. These results confirm that when the stress-strain relation of the second nonlinearly elastic material is convex, the tensile wave can indeed catch the previously transmitted compressive wave. The simulation results by the discontinuous Galerkin method show that the catching-up phenomena can reduce the magnitude of the tensile wave by more than400%. They also show that the non-linear elastic materials have excellent impact resistance.
In the second part, we focus on the impact-induced wave in a slender cylinder composed of a non-convex elastic material. This system is not well-posed due to the non-convexity of the stress-strain relation. A new model equation can be obtained by considering the effects of the radial deformation and traction-free condition on the lateral surface. There are higher order time derivative terms that increase the difficulty for solving. We introduce new time auxiliary variables into the equation so that the model equations can be written as a system with first-order time derivative. Then we present the stability of the local discontinuous Galerkin scheme. The method with pk elements (denoting all polynomials of degree at most k) gives a uniform (k+1)-th order of accuracy forγ in both Ll and L∞norms through one example. In the end, we use local discontinuous Galerkin scheme for a series of simulation, there are some interesting wave patterns, due to the effect of the dispersion and the non-convexity of material, such as the wave pattern with transformation front and solitary wave, the wave pattern with rarefaction wave and solitary wave. We also investigate the interaction of transformation fronts and rarefaction waves, and demonstrate the interesting wave phenomena.
In the third part, we consider the travelling waves in a compressible hyper-elastic rod. The model equation (compressible hyper-elastic rod wave equation) for the ma-terial is similar to the form of Camassa-Holm equation, and there is another constant coefficientλfor the nonlinear term. Therefore, we can refer to the numerical scheme of Camassa-Holm equation for that of this model equation. The Camassa-Holm equation is a completely integrable equation. Unfortunately, the property of completely inte-grable isn't enjoyed by the current model equation when λ≠1. But interestingly, the model equation when λ<0shares traveling waves such as compacton waves, soliton waves. We will focus on these traveling wave, and give numerical simulation for inter-action between these wave.Some unexpected phenomena are found. For example, the compacton and soliton wave are stable with a Gaussian perturbation. After the collision of the two compacton waves, two soliton waves occur. After the collision of the three compactons, three solitons occur. After the collision of compacton wave and soliton wave, the profiles of the finally separated wave from them are likely with that of one soliton wave.
In the fourth part, we continue to consider the wave catching-up phenomena in a nonlinearly elastic composite bar. The solutions contain many shock waves. We use a large-number mesh to capture the information of shock wave, but which unfortunate-ly increases the cost of computation. The discontinuous Galerkin method shares the advantage of easy adaptive implementation since there is no need of continuity on cel-1interface. Therefore, we use adaptive Runge-Kutta discontinuous Galerkin method with KXRCF indicator and TVB indicator to simulate. We consider the effects of the TVB reconstruction, the parameter M, the discontinuous Galerkin scheme with dissi-pation and Pk element, etc. Overall, The results by TVB indicator are better than that by KXRCF indicator, because there is less oscillation. However, there occurs some s-mearing phenomena. Therefore, we need to select an appropriate parameter M to make the results more accurate.
第一部分针对棒状非线性弹性复合材料,其模型方程可以看成是守恒律方程组,但是由于不同材料的(两层)复合,模型方程中流通量项出现间断,且在模型方程的右端源项出现δ函数,这些都给数值格式的设计增加了困难。由于流通量项的不连续性,因此在处理数值流通量的时候需要注意。另外由于物理界面的存在,波发生反射和透射,因此方程的解会包含多个激波,从而也增加了问题的难度。我们首先从物理的角度,引入耗散率和物理容许解的概念,证明间断Galerkin (DG)格式的能量稳定性和DG解是物理容许解。然后借助于耗散率,我们提出一种dissipation-rate reserving DG格式,并通过数值算例比较说明该格式的优点。最后我们利用dissipation-rate reserving DG格式数值模拟了复合材料中波在传播过程中发生碰撞的细节,包括拉伸激波、压缩激波和稀疏波之间的碰撞,并给出了详细的碰撞时间和位置；通过选取四组不同参数比较碰撞之后拉伸波的幅值减小率来说明参数的影响,结果也验证了材料的非线性性能够使得材料中具有破坏性的拉伸波降低幅值达400%。这也表明这样的非线性弹性材料具有很好的抗冲击能力。最后我们考虑应力参数ε以及右边界对波的影响,还有四层复合材料中材料排列方式对材料保护的影响。
第二部分针对细柱状非凸弹性材料,由于应力-应变关系的非凸性,将会导致解的不适定性,因此在材料中考虑柱体径向变形的影响得到新的模型方程,正是这种影响使得新的模型方程中包含高阶时间导数项,从而增加求解的难度。为了便于使用局部间断Galerkin (LDG)方法进行求解,我们引入新的时间辅助变量从而将模型方程转化为只具有一阶的时间导数的方程组,针对这-方程组设计了LDG方法。其次证明了LDG格式的稳定性,并通过数值算例给出LDG格式的精度：在L1模和L∞模意义下应变γ具有k+1阶精度。最后利用LDG格式进行了一系列数值模拟,由于色散和材料非凸性的影响,我们观察到了一些有趣的波动模式(wave pattern),例如,transformation front与类似于孤立子的波组成的波动模式和稀疏波与类似于孤立子的波组成的波动模式等等：另外我们给出了稀疏波与transformation fronts碰撞的详细过程。
第三部分针对杆状可压缩超弹性材料,其模型方程(可压缩超弹性杆波动方程)与Camassa-Holm方程形式很一致,但是在非线性项多一个常数系数γ。正是由于常数系数γ的存在,使得模型方程不再具有好的性质(例如可积性)。模型方程所具有的多种类型波(例如紧孤立波,孤立子波波)是我们重点考察对象,我们利用LDG格式数值模拟这些类型波之间的相互碰撞,发现一些有趣的现象：紧孤立波、孤立子波在Gauss扰动下仍然是稳定的；两个相向的紧孤立波之间碰撞后以及两个右行孤立子波之间碰撞后均有类似于cuspon波形出现；两个紧孤立波碰撞后会出现两个孤立子波；三个紧孤立波碰撞后会出现三个大小不同的孤立子波；一个右行紧孤立波与一个孤立子波碰撞后会出现两个大小不同的孤立子波。
第四部分同样针对棒状非线性弹性复合材料波的传播问题,从第一部分的数值计算结果可以看出应变γ包含多个激波,因此采用较大数目的网格进行模拟来抓住激波的信息,但是网格数目增大同时也大大增加了计算量。因为DG方法具有易于实现自适应的优点,我们采用自适应RKDG方法并结合KXRCF指示子(indicator)和TVB指示子这两种不同的指示子进行模拟,对是否需要TVB重构、参数M、有无耗散DG格式以及Pk元(所有次数不超过k的多项式)等多方面的影响进行考虑。总体而言,TVB指示子的计算结果要比KXRCF指示子好,因为存在的振荡少些。但是激波处出现少许光滑化的现象,因此还需通过选取合适的M值以使得结果更为精确。
In this thesis, we mainly focus on three types of materials:nonlinearly elastic composite bar, non-convex elastic slender cylinder and compressible hyper-elastic rod, and study the wave catching-up and collision phenomena and the phase transition in wave propagation in these materials. We use (local) discontinuous Galerkin method and adaptive Runge-Kutta discontinuous Galerkin method for numerical simulation.
There are four parts in this thesis. In the first part, we consider nonlinearly e-lastic composite bar, in which the model equations is conservation laws. The model equations contain a discontinuous flux and a source term with a δ-function due to the materials'composition. These set difficulties for the design of numerical schemes. Some attention should be paid to deal with the numerical fluxes for the discontinuity of the flux. In addition, the waves can reflect and transmit at the physical interface, and the solutions contain many shock waves, which also increase the difficulty for the problem. From the point of physics, we first introduce the concept of dissipation rate and physically admissible solution. Then, by virtue of the dissipation rate, we develop a dissipation-rate reserving discontinuous Galerkin method and give numerical simula-tions to demonstrate the advantage of the scheme. In the end, we use the dissipation-rate reserving discontinuous Galerkin scheme to simulate the details of the interactions of waves at the propagation process, which include the interactions between tensile shock wave, compressive shock and rarefaction wave. We give the detailed times and po-sitions for the interactions. We also examine how material parameters influence the reduction ratio by four different group of parameters. These results confirm that when the stress-strain relation of the second nonlinearly elastic material is convex, the tensile wave can indeed catch the previously transmitted compressive wave. The simulation results by the discontinuous Galerkin method show that the catching-up phenomena can reduce the magnitude of the tensile wave by more than400%. They also show that the non-linear elastic materials have excellent impact resistance.
In the second part, we focus on the impact-induced wave in a slender cylinder composed of a non-convex elastic material. This system is not well-posed due to the non-convexity of the stress-strain relation. A new model equation can be obtained by considering the effects of the radial deformation and traction-free condition on the lateral surface. There are higher order time derivative terms that increase the difficulty for solving. We introduce new time auxiliary variables into the equation so that the model equations can be written as a system with first-order time derivative. Then we present the stability of the local discontinuous Galerkin scheme. The method with pk elements (denoting all polynomials of degree at most k) gives a uniform (k+1)-th order of accuracy forγ in both Ll and L∞norms through one example. In the end, we use local discontinuous Galerkin scheme for a series of simulation, there are some interesting wave patterns, due to the effect of the dispersion and the non-convexity of material, such as the wave pattern with transformation front and solitary wave, the wave pattern with rarefaction wave and solitary wave. We also investigate the interaction of transformation fronts and rarefaction waves, and demonstrate the interesting wave phenomena.
In the third part, we consider the travelling waves in a compressible hyper-elastic rod. The model equation (compressible hyper-elastic rod wave equation) for the ma-terial is similar to the form of Camassa-Holm equation, and there is another constant coefficientλfor the nonlinear term. Therefore, we can refer to the numerical scheme of Camassa-Holm equation for that of this model equation. The Camassa-Holm equation is a completely integrable equation. Unfortunately, the property of completely inte-grable isn't enjoyed by the current model equation when λ≠1. But interestingly, the model equation when λ<0shares traveling waves such as compacton waves, soliton waves. We will focus on these traveling wave, and give numerical simulation for inter-action between these wave.Some unexpected phenomena are found. For example, the compacton and soliton wave are stable with a Gaussian perturbation. After the collision of the two compacton waves, two soliton waves occur. After the collision of the three compactons, three solitons occur. After the collision of compacton wave and soliton wave, the profiles of the finally separated wave from them are likely with that of one soliton wave.
In the fourth part, we continue to consider the wave catching-up phenomena in a nonlinearly elastic composite bar. The solutions contain many shock waves. We use a large-number mesh to capture the information of shock wave, but which unfortunate-ly increases the cost of computation. The discontinuous Galerkin method shares the advantage of easy adaptive implementation since there is no need of continuity on cel-1interface. Therefore, we use adaptive Runge-Kutta discontinuous Galerkin method with KXRCF indicator and TVB indicator to simulate. We consider the effects of the TVB reconstruction, the parameter M, the discontinuous Galerkin scheme with dissi-pation and Pk element, etc. Overall, The results by TVB indicator are better than that by KXRCF indicator, because there is less oscillation. However, there occurs some s-mearing phenomena. Therefore, we need to select an appropriate parameter M to make the results more accurate.
引文
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