高超声速稀薄气流非结构网格DSMC及并行算法研究
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摘要
高超声速飞行器在高空飞行时,随着飞行高度的增加,流场的稀薄气体效应随之增大,此时已不再能用连续介质的方法研究高空高速气流中所发生的各种现象。受类似条件下地面实验设备及相关测试技术所限,求解Boltzmann方程又存在着巨大的困难,直接模拟Monte Carlo方法(DSMC)已经成为非常重要的研究手段,也是求解稀薄气体动力学问题唯一获得巨大成功的方法。本文主要目的是通过深入研究高超声速稀薄气体流动的非结构网格直接模拟Monte Carlo方法及其并行计算技术,致力于开发一套适用于任意复杂外形的高效通用DSMC模拟程序。
首先,研究了直接模拟Monte Carlo方法所必需的基本分子气体动力学理论,为流场的直接物理模拟打下理论基础。论述了二元弹性碰撞中的力学机理并推导了分子碰撞后速度的求解公式。讨论了分子间的作用势与分子碰撞模型。在前人工作的基础上,以Lennard-Jones势为作用模型,设计了GSS-3分子碰撞模型,该模型同时具有吸引和排斥的分子作用力,能够给出正确的气体输运性质。
其次,研究了高超声速稀薄气流非结构网格DSMC方法的实现策略及应用。将Bird位置元方案中的子网格思想引入到非结构网格上来,只存储子网格的总体标识号,利用较少的计算网格又能提高分子的分辨率和计算精度。提出了将面积元/体积元坐标搜索技术与交替数字二叉树搜索算法(ADT)相结合来跟踪模拟分子在网格之间的迁移,使用ADT方法判别分子与物面是否相交,节约了计算时间。同时编制了相应的计算程序,对具体的算例进行了数值模拟。
然后,在非结构网格DSMC方法实现的基础上,对其中的关键技术进行了优化与设计。以面积元/体积元坐标搜索技术为原型,设计了一类跟踪模拟分子迁移的新算法,该算法仅需少量的逻辑运算与代数运算,较之以往需要大量的逻辑运算与浮点运算节约了计算时间,同时新算法不仅可以跟踪模拟分子在网格之间的迁移,而且可以准确判别分子与物面是否相互作用,搜索过程中的附带信息给出了分子与物面碰撞的确切时间与精确位置。为了加快流场的时间发展历程,提出了适合非结构网格DSMC方法的动态局部时间步长技术,该项技术的应用能够较大幅度地缩短流场模拟的计算时间。在非结构网格DSMC中,引入了碰撞距离的思想,可以有效避免因网格尺度过大导致出现严重的计算误差。同时,本文对传统DSMC方法中的数据结构进行了优化,设计了局部化的数据组织方式,节约了内存,变间接访问数据为直接访问,节约了计算时间。在程序编制过程中,充分展现了Fortran90高级语言的主要特性,引人动态数组、指针、链表以及派生类型数据。最后对过渡领域高超声速绕流进行了数值模拟并对计算效率进行了比较。
接下来,研究了化学反应气体流动的非结构网格DSMC及其应用。论述了碰撞过程中的化学反应理论,给出了不同分子碰撞模型下的化学反应抽样几率函数(位阻因子)构造的过程。同时编制了化学反应气体流动非结构网格DSMC的计算程序,给出了具体的数值算例。
最后,基于PC-CLUSTER群机并行体系结构与消息传递库MPI环境开展了非结构网格DSMC并行计算技术的研究。提出了一类基于结构背景网格上的动态非结构网格分区策略,保证各子区域的分子数大致相等,实现计算进程间的动态负载平衡。利用MPI库函数构造了两类符合DSMC并行原理的通信方法:单步通讯法与多步通讯法。采用单控制多数据流(SPMD)以及Master/Slave并行模式,设计了非结构网格DSMC整体并行算法。最后对高超声速绕流进行并行计算,取得了较为理想的加速比与计算效率。
Hypersonic vehicles encounter a broad range of flight conditions varying from continuum to rarefied flows. In rarefied regime, the molecular effects are important and the assumptions of continuum methods become invalid. Due to the facts that the available ground tunnel experimental techniques and test facilities are limited and attempts to solve the Boltzmann equation have met with many difficulties, the Direct Simulation Monte Carlo (DSMC), originally developed by Bird in 1960’s, has become a widely used computational tool for the rarefied gas flows. The purpose of the present thesis is to study detailed DSMC method on unstructured grids for hypersonic rarefied gas flows and to develop highly efficient automated DSMC software that is suitable for any complex geometry.
First of all, necessary basic molecular kinetic theory of gases is studied for DSMC. Binary elastic collision is analyzed in detail and the expression of post collision velocities is deduced. Intermolecular potential and different collision model is discussed. Based on some existing results, a new molecular collision model for DSMC, called GSS-3 model, is introduced to provide consistency for the transport properties with Lennard-Jones potentials that is a realistic intermolecular potential.
Secondly, the implementation of DSMC method on unstructured grids is investigated to calculate hypersonic rarefied flow. The idea of sub-cell in Bird’s Position Element Method is introduced into the DSMC method on unstructured grids, by which only ID number of sub-cell is stored. An automatic searching method is presented to improve efficiency and save running time, which is a coupling of area/volume coordinate searching technique and alternative digital tree (ADT) searching algorithm. As a consequence, not only is the computation time reduced, but also the precision is improved largely. The deterministic criterion for a molecule to reflect on a certain surface element is used by ADT algorithm instead of a probabilistic one. Extensive numerical experiments are made to directly simulate hypersonic rarefied gas flows.
Then, based on the work of implementation of DSMC method on unstructured grids, this thesis is to study strategies for optimization and improvement of key technique in DSMC method on unstructured grids. A new automatic searching algorithm is presented by calculating intersection point, by which all information about molecules hitting surface boundary can be given and the deterministic criterion is used for a molecule to reflect on a certain surface element instead of a probabilistic one as used in position element method. In order to improve efficiency and save running time, an adaptive local time stepping suitable for DSMC method on unstructured grids is designed. Collision distance is introduced to DSMC method on unstructured grids which can avoid bad errors due to large scale dimension of some unstructured grids. A new data structure based on unstructured grids is introduced to achieve high performance and efficiency. Accessing molecular data is direct other than traditional data structure that is indirect by cross-reference array. Dynamic allocation feature of Fortran90 is fully exploited, which makes the code more flexible. By comparisons of efficiencies of numerical experiments, the feasibility of the method is confirmed.
Afterwards, DSMC method on unstructured grids is studied to calculate rarefied gas flows with chemical reactions. The collision theory for chemical reactions is discussed and the probability function of reaction (or steric factor) of different collision models is deduced for direct simulation of chemical gas flow. A subroutine is incorporated into existing DSMC code on unstructured grids in order to simulate chemical reaction rarefied gas flow.
At last, this thesis is to investigate parallel algorithm of DSMC method on unstructur- ed grids using the MPI standard library on PC-CLUSTER that is distributed memory architecture. In order to obtain high parallel performance, a new adaptive domain decomposition technique is presented. Because this method results in equal number of molecules among processors, dynamic load balance is maintained. Two different communication methods are constructed by means of MPI library: single step communication and multiple step communication method. A complete parallel implementation of DSMC method on unstructured grids included 2D and 3D is developed using SPMD and master/slave parallel mode. This new tool is applied to numerical experiments, which are used to evaluate the performance of parallel DSMC code on unstructured grids.
首先,研究了直接模拟Monte Carlo方法所必需的基本分子气体动力学理论,为流场的直接物理模拟打下理论基础。论述了二元弹性碰撞中的力学机理并推导了分子碰撞后速度的求解公式。讨论了分子间的作用势与分子碰撞模型。在前人工作的基础上,以Lennard-Jones势为作用模型,设计了GSS-3分子碰撞模型,该模型同时具有吸引和排斥的分子作用力,能够给出正确的气体输运性质。
其次,研究了高超声速稀薄气流非结构网格DSMC方法的实现策略及应用。将Bird位置元方案中的子网格思想引入到非结构网格上来,只存储子网格的总体标识号,利用较少的计算网格又能提高分子的分辨率和计算精度。提出了将面积元/体积元坐标搜索技术与交替数字二叉树搜索算法(ADT)相结合来跟踪模拟分子在网格之间的迁移,使用ADT方法判别分子与物面是否相交,节约了计算时间。同时编制了相应的计算程序,对具体的算例进行了数值模拟。
然后,在非结构网格DSMC方法实现的基础上,对其中的关键技术进行了优化与设计。以面积元/体积元坐标搜索技术为原型,设计了一类跟踪模拟分子迁移的新算法,该算法仅需少量的逻辑运算与代数运算,较之以往需要大量的逻辑运算与浮点运算节约了计算时间,同时新算法不仅可以跟踪模拟分子在网格之间的迁移,而且可以准确判别分子与物面是否相互作用,搜索过程中的附带信息给出了分子与物面碰撞的确切时间与精确位置。为了加快流场的时间发展历程,提出了适合非结构网格DSMC方法的动态局部时间步长技术,该项技术的应用能够较大幅度地缩短流场模拟的计算时间。在非结构网格DSMC中,引入了碰撞距离的思想,可以有效避免因网格尺度过大导致出现严重的计算误差。同时,本文对传统DSMC方法中的数据结构进行了优化,设计了局部化的数据组织方式,节约了内存,变间接访问数据为直接访问,节约了计算时间。在程序编制过程中,充分展现了Fortran90高级语言的主要特性,引人动态数组、指针、链表以及派生类型数据。最后对过渡领域高超声速绕流进行了数值模拟并对计算效率进行了比较。
接下来,研究了化学反应气体流动的非结构网格DSMC及其应用。论述了碰撞过程中的化学反应理论,给出了不同分子碰撞模型下的化学反应抽样几率函数(位阻因子)构造的过程。同时编制了化学反应气体流动非结构网格DSMC的计算程序,给出了具体的数值算例。
最后,基于PC-CLUSTER群机并行体系结构与消息传递库MPI环境开展了非结构网格DSMC并行计算技术的研究。提出了一类基于结构背景网格上的动态非结构网格分区策略,保证各子区域的分子数大致相等,实现计算进程间的动态负载平衡。利用MPI库函数构造了两类符合DSMC并行原理的通信方法:单步通讯法与多步通讯法。采用单控制多数据流(SPMD)以及Master/Slave并行模式,设计了非结构网格DSMC整体并行算法。最后对高超声速绕流进行并行计算,取得了较为理想的加速比与计算效率。
Hypersonic vehicles encounter a broad range of flight conditions varying from continuum to rarefied flows. In rarefied regime, the molecular effects are important and the assumptions of continuum methods become invalid. Due to the facts that the available ground tunnel experimental techniques and test facilities are limited and attempts to solve the Boltzmann equation have met with many difficulties, the Direct Simulation Monte Carlo (DSMC), originally developed by Bird in 1960’s, has become a widely used computational tool for the rarefied gas flows. The purpose of the present thesis is to study detailed DSMC method on unstructured grids for hypersonic rarefied gas flows and to develop highly efficient automated DSMC software that is suitable for any complex geometry.
First of all, necessary basic molecular kinetic theory of gases is studied for DSMC. Binary elastic collision is analyzed in detail and the expression of post collision velocities is deduced. Intermolecular potential and different collision model is discussed. Based on some existing results, a new molecular collision model for DSMC, called GSS-3 model, is introduced to provide consistency for the transport properties with Lennard-Jones potentials that is a realistic intermolecular potential.
Secondly, the implementation of DSMC method on unstructured grids is investigated to calculate hypersonic rarefied flow. The idea of sub-cell in Bird’s Position Element Method is introduced into the DSMC method on unstructured grids, by which only ID number of sub-cell is stored. An automatic searching method is presented to improve efficiency and save running time, which is a coupling of area/volume coordinate searching technique and alternative digital tree (ADT) searching algorithm. As a consequence, not only is the computation time reduced, but also the precision is improved largely. The deterministic criterion for a molecule to reflect on a certain surface element is used by ADT algorithm instead of a probabilistic one. Extensive numerical experiments are made to directly simulate hypersonic rarefied gas flows.
Then, based on the work of implementation of DSMC method on unstructured grids, this thesis is to study strategies for optimization and improvement of key technique in DSMC method on unstructured grids. A new automatic searching algorithm is presented by calculating intersection point, by which all information about molecules hitting surface boundary can be given and the deterministic criterion is used for a molecule to reflect on a certain surface element instead of a probabilistic one as used in position element method. In order to improve efficiency and save running time, an adaptive local time stepping suitable for DSMC method on unstructured grids is designed. Collision distance is introduced to DSMC method on unstructured grids which can avoid bad errors due to large scale dimension of some unstructured grids. A new data structure based on unstructured grids is introduced to achieve high performance and efficiency. Accessing molecular data is direct other than traditional data structure that is indirect by cross-reference array. Dynamic allocation feature of Fortran90 is fully exploited, which makes the code more flexible. By comparisons of efficiencies of numerical experiments, the feasibility of the method is confirmed.
Afterwards, DSMC method on unstructured grids is studied to calculate rarefied gas flows with chemical reactions. The collision theory for chemical reactions is discussed and the probability function of reaction (or steric factor) of different collision models is deduced for direct simulation of chemical gas flow. A subroutine is incorporated into existing DSMC code on unstructured grids in order to simulate chemical reaction rarefied gas flow.
At last, this thesis is to investigate parallel algorithm of DSMC method on unstructur- ed grids using the MPI standard library on PC-CLUSTER that is distributed memory architecture. In order to obtain high parallel performance, a new adaptive domain decomposition technique is presented. Because this method results in equal number of molecules among processors, dynamic load balance is maintained. Two different communication methods are constructed by means of MPI library: single step communication and multiple step communication method. A complete parallel implementation of DSMC method on unstructured grids included 2D and 3D is developed using SPMD and master/slave parallel mode. This new tool is applied to numerical experiments, which are used to evaluate the performance of parallel DSMC code on unstructured grids.
引文
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