基于并行的特大增量步算法在计算固体力学中的应用
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摘要
广义逆力法是一种基于加载全过程的迭代算法，因解决材料非线性问题时增量步长不受计算精度限制只由加卸载状态决定，又称特大增量步算法（Large Increment Method,简称LIM）。作为一种有限元算法，LIM具有很强的空间上的可并行性，无需进行子结构划分就可以直接以单元为最小单位并行求解本构方程。由于引入了整体迭代优化算法，在分析材料非线性问题时，LIM具有很强的时间上的可并行性，这是这一算法独具的优点。在复杂的加载情况下，时间上的并行可大大提高计算速度。本文分析了本算法的时间并行性优势，找出需要进一步完善的算法并行化的前期工作和难点，提出了包含数值并行、空间并行及时间并行的算法并行化解决方案。但由于并行计算是应对大规模计算需求的产物，计算规模越大，并行越有效。而在本文的工作开展之前，LIM由于单元库贫乏，只能应对杆系结构的分析及简单的二维问题的计算，无法应对复杂结构的分析，计算规模无法增大。因而当务之急是丰满单元库。当算法无论从计算精度还是计算规模上都满足实用要求，再进行算法的并行化，将能充分发挥算法的并行特点，大大提高计算效率。
本文的主要工作着重于LIM在计算固体力学中的应用扩展——即开发二维和三维实体单元、构建LIM单元库。论文首先将杆系单元写入LIM的串行程序单元库，以其为例讨论并突出了LIM独有的计算特点。为丰满单元库，文中创新性地提出了两种建立二维和三维实体单元的通用办法，建立了两类单元，其一命名为“节点力型”单元，另一种命名为“应力型”单元。
“节点力型”单元是选取单元部分节点力作为单元广义内力的一类单元。单元广义内力由能够唯一确定单元内部应力分布的最少独立未知力组成，可以有多种选择。在综合考虑了各种二维、三维单元之后，本文提出了选取单元基本静定体系的特殊方式，确定了基本静定体系的局部坐标并选取了合理的基本力系作为单元广义内力系。在这样明确定义的基本体系下，文中通过坐标转换及平衡分析建立了单元的控制方程通式。通式没有限定单元的形状与节点个数，给定节点数目即可通过二维或三维通式建立多节点单元的控制方程。而且通式建立过程系统规范，不存在随机性，非常利于编程计算。本文已将四个二维及两个三维“节点力型”单元增加至LIM单元库，并在LIM串行程序中完成了单元控制方程通式及上述各单元的编写工作。数值算例表明，使用“节点力型”单元时，LIM的计算收敛速度及计算精度满足使用要求，精度受一定畸变度的不规则单元影响小，数值计算不受全局坐标系取向及节点编号的影响。
“应力型”单元分别采用不同的插值函数逼近应力场和位移场，再通过虚功原理建立单元的控制方程。由于将应力场插值函数的线性无关系数定义为单元广义内力，因而命名为“应力型”。本文首先通过假设应力函数得到应力各分量表达式，并建立了二维及三维单元的单元控制方程；而后，基于LIM的控制方程，文中提出了判别零能变形模式的充要条件，并分别从单元柔度矩阵是否病态以及是否存在零能变形模式两个方面考查了各单元。为抑制零能变形模式，论文提出了另一种构造“应力型”单元的方法，即基于基本位移模式的等函数法，并利用等函数法构造了一个新的无零能变形模式的六面体单元。基于以上两种方法，本文为LIM单元库增加了六个二维及三个三维“应力型”单元，并完成了所有单元的程序编写工作。数值算例表明，使用“应力型”单元时，LIM的计算收敛速度及计算精度均满足使用要求。
为了便于用户使用LIM求解弹塑性问题，文章在最后提出了适用于LIM的一维至三维一致弹塑性矩阵的推导，并以数值算例展示了LIM在求解弹塑性问题中的适用性及并行潜能。
本文共为LIM算法增加二维平面实体单元十个、三维空间实体单元五个。建立单元的方法具有可扩展性，用户可以根据需要、依据文中提出的方法增加适用单元。论文研究成果使得一直停滞不前的LIM的研究向“在计算固体力学中的应用”方面迈进了一大步。同时，论文作者独立完成了LIM的面向对象的串行程序的全部编写工作。在此基础上，对程序的进一步优化工作以及实现并行化计算将可以顺利开展。
General Inverse Force Method is an iteration procedure covering whole loading path. Itsincrement steps are determined by the changes of loading statement. Unlike the traditionalFinite Element Method, the size of its each increment step is not restricted to assure precision,thus it is often called Large Increment Method (LIM). As a finite element algorithm, LIM wasborn with the capacity for numerical parallel computation and the parallel computation inspatial domain. Better still, it can parallely solve the constitutive equations of each elementwithout dividing a structure into several substructures. Moreover, LIM was endowed with theadvantage of the parallel computation also in time domain, which is not provided by any otherfinite element algorithms. In the case that the loading path is complicated, the parallelcomputing in time domain ensures the potential of greatly accelerating computation speed. Inthe first part of the present dissertation, the parallel computation in time domain washighlighted; the deficiencies and difficulties of the preliminary research of the parallelizationfor LIM were pointed out; and two parallel Scheduling Schema including numerical parallelcomputation as well as parallel computations in spatial and time domain were provided. As isknown to all, parallel computing was proposed to deal with the large scale computations. Thelager the scale is, the higher the computational efficiency could be. However, before thispaper, LIM had no capability to cope with the analysis of complex solid structure whichrequires large scale computation, because of the insufficiency of element types. Therefore, themost important and prerequisite thing is to construct an element library for LIM, and make itrich. Only when LIM can cope with the large scale computational problems and provideprecise computational results, the parallelization of LIM can be implemented efficiently, andthe power of parallel computation for LIM can be shown.
In this dissertation, the key work was the extended application of LIM in computationalsolid mechanics. Detailedly speaking, the key work was to develop2D and3D solid elements,and then to construct a finite element library for LIM. First of all,3D frame element wasabsorbed into the element library and coded for LIM program. The computing features ofLIM were emphasized based on the discussion of the numerical computation using frame element. To enrich the finite element library, two methods for developing2D and3D solidelements were proposed creatively. Accordingly, two types of elements were proposed, onewas named 'nodal force type' element and the other was named 'stress type' element.
Nodal force type element was named by the reason that parts of the elemental nodalforces were employed as the elemental generalized inner forces. Generally speaking, theelemental generalized inner forces were composed of the fewest independent unknown forceswhich are sufficient to determine the stress distribution within an element, and the unknownforces could be chosen arbitrarily. Therefore, in consideration of all kinds of2D and3Delements, the special elemental basic statically determinate system was chosen, the localcoordinates of this system was defined. Hence, the basic force system was chosen, and theappropriate elemental generalized inner forces were determined. Under the clearly definedbasic system, the general forms of the elemental governing equations were proposed byconsidering coordinates transformation and equilibrium relationship. Because that the shapeof element and the number of nodes were not limited in the general forms of the elementalgoverning equations, one can obtain any2D or3D solid element with a given number ofnodes by implementing the proposed general forms. It should be noted that the general formsof the elemental governing equations can be easily realized by programming because of itsstrong systematicness. In the present dissertation, four2D elements and two3D elementswere developed for the element library of LIM, and the codes of the proposed general formsand elements were accomplished. Some illustrative numerical examples were solved using theproposed library. The results were compared with those obtained from displacement-basedfinite element method and analytical close-form solutions, which has clearly shown thecomputational convergence rate and accuracy of LIM. Furthermore, the insensitiveness to themesh distortion, coordinates as well as the sequence of mesh node labels was also observed.
The governing equations of stress type element were obtained by introducing principle ofvirtual work. With in an element, the approximations of stress field and displacement fieldwere presented as interpolation polynomials, respectively. Because that the linear independentcoefficients of the assumed interpolation polynomials of stress field were employed aselemental generalized inner forces, the elements were named stress type elements. In thepresent dissertation, the polynomial of each stress component was deduced from the assumed stress function firstly, and then the elemental governing equations were obtained,consequently the stress type elements were developed. Based on the elemental governingequations, the necessary and sufficient condition for determination of spurious zero energymodes is presented, and the spurious zero energy modes as well as the ill conditionedflexibility matrix were detected to avoid numerical error. To suppress spurious zero energymodes, another method named iso-function method which was based on basic displacementmodes was employed. By using which, one hexahedral element with the absence of spuriouszero energy modes was developed. In two manners described above, six2D elements andthree3D elements were developed for the element library, and the codes of the proposedelements were completed. The illustrative numerical examples were also solved using theproposed stress type library and the computational convergence rate and accuracy of LIM wasshown.
To make LIM more practical in solving material nonlinearity problem, the consistentelastoplastic matrix for elemental stage is provided in the last part of the present dissertation.By the numerical example, the applicability of LIM was proved, and the potential of parallelcomputation was imply.
In summary, the solid element library which including ten2D elements and five3Delements was proposed for LIM modeling arbitrary configurations. Because of theextensibility of the proposed manners for obtaining elements, following the methods, userscan develop their own elements as needed. Benefit from the achievements of the presentdissertation, the application of LIM in computational solid mechanics has made a remarkableprogress. Meanwhile, the object oriented serial program of LIM has been completed in themain, based upon which, the optimization and parallelization of the program can be done inthe future.
本文的主要工作着重于LIM在计算固体力学中的应用扩展——即开发二维和三维实体单元、构建LIM单元库。论文首先将杆系单元写入LIM的串行程序单元库，以其为例讨论并突出了LIM独有的计算特点。为丰满单元库，文中创新性地提出了两种建立二维和三维实体单元的通用办法，建立了两类单元，其一命名为“节点力型”单元，另一种命名为“应力型”单元。
“节点力型”单元是选取单元部分节点力作为单元广义内力的一类单元。单元广义内力由能够唯一确定单元内部应力分布的最少独立未知力组成，可以有多种选择。在综合考虑了各种二维、三维单元之后，本文提出了选取单元基本静定体系的特殊方式，确定了基本静定体系的局部坐标并选取了合理的基本力系作为单元广义内力系。在这样明确定义的基本体系下，文中通过坐标转换及平衡分析建立了单元的控制方程通式。通式没有限定单元的形状与节点个数，给定节点数目即可通过二维或三维通式建立多节点单元的控制方程。而且通式建立过程系统规范，不存在随机性，非常利于编程计算。本文已将四个二维及两个三维“节点力型”单元增加至LIM单元库，并在LIM串行程序中完成了单元控制方程通式及上述各单元的编写工作。数值算例表明，使用“节点力型”单元时，LIM的计算收敛速度及计算精度满足使用要求，精度受一定畸变度的不规则单元影响小，数值计算不受全局坐标系取向及节点编号的影响。
“应力型”单元分别采用不同的插值函数逼近应力场和位移场，再通过虚功原理建立单元的控制方程。由于将应力场插值函数的线性无关系数定义为单元广义内力，因而命名为“应力型”。本文首先通过假设应力函数得到应力各分量表达式，并建立了二维及三维单元的单元控制方程；而后，基于LIM的控制方程，文中提出了判别零能变形模式的充要条件，并分别从单元柔度矩阵是否病态以及是否存在零能变形模式两个方面考查了各单元。为抑制零能变形模式，论文提出了另一种构造“应力型”单元的方法，即基于基本位移模式的等函数法，并利用等函数法构造了一个新的无零能变形模式的六面体单元。基于以上两种方法，本文为LIM单元库增加了六个二维及三个三维“应力型”单元，并完成了所有单元的程序编写工作。数值算例表明，使用“应力型”单元时，LIM的计算收敛速度及计算精度均满足使用要求。
为了便于用户使用LIM求解弹塑性问题，文章在最后提出了适用于LIM的一维至三维一致弹塑性矩阵的推导，并以数值算例展示了LIM在求解弹塑性问题中的适用性及并行潜能。
本文共为LIM算法增加二维平面实体单元十个、三维空间实体单元五个。建立单元的方法具有可扩展性，用户可以根据需要、依据文中提出的方法增加适用单元。论文研究成果使得一直停滞不前的LIM的研究向“在计算固体力学中的应用”方面迈进了一大步。同时，论文作者独立完成了LIM的面向对象的串行程序的全部编写工作。在此基础上，对程序的进一步优化工作以及实现并行化计算将可以顺利开展。
General Inverse Force Method is an iteration procedure covering whole loading path. Itsincrement steps are determined by the changes of loading statement. Unlike the traditionalFinite Element Method, the size of its each increment step is not restricted to assure precision,thus it is often called Large Increment Method (LIM). As a finite element algorithm, LIM wasborn with the capacity for numerical parallel computation and the parallel computation inspatial domain. Better still, it can parallely solve the constitutive equations of each elementwithout dividing a structure into several substructures. Moreover, LIM was endowed with theadvantage of the parallel computation also in time domain, which is not provided by any otherfinite element algorithms. In the case that the loading path is complicated, the parallelcomputing in time domain ensures the potential of greatly accelerating computation speed. Inthe first part of the present dissertation, the parallel computation in time domain washighlighted; the deficiencies and difficulties of the preliminary research of the parallelizationfor LIM were pointed out; and two parallel Scheduling Schema including numerical parallelcomputation as well as parallel computations in spatial and time domain were provided. As isknown to all, parallel computing was proposed to deal with the large scale computations. Thelager the scale is, the higher the computational efficiency could be. However, before thispaper, LIM had no capability to cope with the analysis of complex solid structure whichrequires large scale computation, because of the insufficiency of element types. Therefore, themost important and prerequisite thing is to construct an element library for LIM, and make itrich. Only when LIM can cope with the large scale computational problems and provideprecise computational results, the parallelization of LIM can be implemented efficiently, andthe power of parallel computation for LIM can be shown.
In this dissertation, the key work was the extended application of LIM in computationalsolid mechanics. Detailedly speaking, the key work was to develop2D and3D solid elements,and then to construct a finite element library for LIM. First of all,3D frame element wasabsorbed into the element library and coded for LIM program. The computing features ofLIM were emphasized based on the discussion of the numerical computation using frame element. To enrich the finite element library, two methods for developing2D and3D solidelements were proposed creatively. Accordingly, two types of elements were proposed, onewas named 'nodal force type' element and the other was named 'stress type' element.
Nodal force type element was named by the reason that parts of the elemental nodalforces were employed as the elemental generalized inner forces. Generally speaking, theelemental generalized inner forces were composed of the fewest independent unknown forceswhich are sufficient to determine the stress distribution within an element, and the unknownforces could be chosen arbitrarily. Therefore, in consideration of all kinds of2D and3Delements, the special elemental basic statically determinate system was chosen, the localcoordinates of this system was defined. Hence, the basic force system was chosen, and theappropriate elemental generalized inner forces were determined. Under the clearly definedbasic system, the general forms of the elemental governing equations were proposed byconsidering coordinates transformation and equilibrium relationship. Because that the shapeof element and the number of nodes were not limited in the general forms of the elementalgoverning equations, one can obtain any2D or3D solid element with a given number ofnodes by implementing the proposed general forms. It should be noted that the general formsof the elemental governing equations can be easily realized by programming because of itsstrong systematicness. In the present dissertation, four2D elements and two3D elementswere developed for the element library of LIM, and the codes of the proposed general formsand elements were accomplished. Some illustrative numerical examples were solved using theproposed library. The results were compared with those obtained from displacement-basedfinite element method and analytical close-form solutions, which has clearly shown thecomputational convergence rate and accuracy of LIM. Furthermore, the insensitiveness to themesh distortion, coordinates as well as the sequence of mesh node labels was also observed.
The governing equations of stress type element were obtained by introducing principle ofvirtual work. With in an element, the approximations of stress field and displacement fieldwere presented as interpolation polynomials, respectively. Because that the linear independentcoefficients of the assumed interpolation polynomials of stress field were employed aselemental generalized inner forces, the elements were named stress type elements. In thepresent dissertation, the polynomial of each stress component was deduced from the assumed stress function firstly, and then the elemental governing equations were obtained,consequently the stress type elements were developed. Based on the elemental governingequations, the necessary and sufficient condition for determination of spurious zero energymodes is presented, and the spurious zero energy modes as well as the ill conditionedflexibility matrix were detected to avoid numerical error. To suppress spurious zero energymodes, another method named iso-function method which was based on basic displacementmodes was employed. By using which, one hexahedral element with the absence of spuriouszero energy modes was developed. In two manners described above, six2D elements andthree3D elements were developed for the element library, and the codes of the proposedelements were completed. The illustrative numerical examples were also solved using theproposed stress type library and the computational convergence rate and accuracy of LIM wasshown.
To make LIM more practical in solving material nonlinearity problem, the consistentelastoplastic matrix for elemental stage is provided in the last part of the present dissertation.By the numerical example, the applicability of LIM was proved, and the potential of parallelcomputation was imply.
In summary, the solid element library which including ten2D elements and five3Delements was proposed for LIM modeling arbitrary configurations. Because of theextensibility of the proposed manners for obtaining elements, following the methods, userscan develop their own elements as needed. Benefit from the achievements of the presentdissertation, the application of LIM in computational solid mechanics has made a remarkableprogress. Meanwhile, the object oriented serial program of LIM has been completed in themain, based upon which, the optimization and parallelization of the program can be done inthe future.
引文
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