一种面向车身结构设计的大规模问题快速计算方法
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摘要
车身结构设计问题涉及到对复杂结构在多参数条件下多次进行大规模问题求解,是一个较为耗时的过程。本文从节约计算成本和提高计算精度的角度提出一种面向车身设计的复杂结构多参数大规模问题快速计算方法对车身结构进行设计。
快速计算方法的基本思路是当系统由多个参数来描述时,这些参数组合会使系统有不同的响应,这些响应可以由有限次试验得到,而系统在新参数下的解可以用事先设计的近似解空间来表示。在实际计算过程中,应用迦辽金映射技术,将结构设计问题向一个事先构造的子空间中进行投影,将大规模问题缩减为小规模问题,该小规模问题即可快速求解。这种方法在对结构进行自由度缩减时能很好地保持原系统的物理属性。论文从以下方面展开工作。
第一,从壳单元的有限元格式入手,对壳单元刚度矩阵在单元水平上进行参数化分解,同时,按照设计问题的具体要求,将具有同一参数特征的单元刚度矩阵按照有限元方法的基本原理进行坐标转换和组装,最终将整体刚度矩阵表达为一种显式参数化格式。显式参数化格式保持了设计问题的参数特性。
第二,提出一种新的刚度矩阵存储技术,将总体刚度矩阵按照节点的物理属性进行存储,大幅节约了存储空间。对单元刚度矩阵的组成和物理意义进行分析,将单元刚度矩阵按照节点大小的顺序排列成六维的矩阵形式；整体坐标系下的刚度矩阵的六维存储格式通过对单元刚度矩阵在对应位置上的组装实现。在六维格式的基础上,对六维格式的刚度矩阵进行Cholesky分解,并求解线性方程组；考虑到在快速计算方法中需要对矩阵进行的投影运算,对六维存储格式的刚度矩阵进行投影运算研究。
第三,提出一种直接正交构造子空间的方法,将该方法应用于某车身结构的快速计算中,评价计算的精度和计算成本。直接正交法在进行子空间构造时,首先使用奇异值分解对样本点的有限元解进行正交归一化,左奇异向量即为所构造的近似子空间的一个基。直接正交法比传统的贪婪算法在计算成本方面具有较大的优势。
第四,为了进一步提高快速计算方法的计算效率,提出一种分级自适应方法。该方法以直接正交法和贪婪算法为基础,将构造子空间的过程分为初级子空间构造过程和最终子空间构造过程。在初级子空间构造过程中使用直接正交法进行计算,以节省计算量为主要目的,在子空间维数较少的情况下,直接正交法的数值损失较小；在初级子空间的基础上使用贪婪算法进行最终子空间的构造,通过误差控制保证所构造子空间的精度。同时,对初始子空间的维数对最终计算的精度和计算量的影响进行了研究。
第五,针对快速计算方法的较为复杂的计算流程,提出一种自动快速计算方法,该方法通过参数域自动离散、有限元计算、子空间构造和子空间实时更新技术实现了快速计算方法的自动化。通过自动快速计算方法的实现,改变了传统的基于减基法基本思想的快速计算方法的基本计算流程和误差评价方法,形成了一种实时构造、实时更新、实时误差评价的新的快速计算方法。
在研究过程中,使用多个车身设计算例对提出的方法进行验证,结果表明,文中提出的快速计算方法在精度和效率上都比常规的减基法有较大的提高。文中提出的方法在车身设计问题中具有较高的计算效率,并可以将其应用于其它的设计工况中,特别适用于有实时计算要求的场合。对于工程实际中的壳结构优化问题,使用文中提出的方法可以快速地得到最优解。同时,通过改变参数化有限元格式,文中提出的方法同样可以用于梁结构以及实体结构等的设计中。
Vehicle body design phase is a time-consuming process which requires the solution of a large-scale problem with many parameters. A new computational method is proposed in the paper to achieve a rapid way of solving the large-scale linear algebraic equations with many parameters in vehicle design process with respect to the computational cost and the computational accuracy.
The basic idea of the method is that although the field variable generally belongs to the infinite-dimensional space associated with the underlying partial differential equation, indeed, it resides on a low-dimensional subspace induced by the parametric dependence. In computational practice, the large-scale system is reduced to a smaller one by Galerkin projection based on that subspace. The structural behaviors are predicted using the reduced system. The physical property of the problem is retained during the projection.The research is carried out in following sections.
First, the finite element formulation of shell element is analyzed and the element stiffness matrix is decomposed to achieve a parametrized formulation on the element level. Then the element matrices with the same parametric properties are assembled based on the finite element method via coordinate transforming. The parametric property is retained in the form of the explicit parametrized finite element formulation.
Second, a new storage method of large-scale matrix named six-dimensional method is proposed to save the computational cost and improve the computational accuracy. In the new storage formulation, the global stiffness matrix is stored according to the node information which indicates the physical properties of each node. The components of the element stiffness matrices are analyzed first and the locations of the element stiffness matrices in six-dimensional matrix are determined according to the order of the nodes. The global stiffness matrix in six-dimensional form is achieved by assembling the element stiffness in the corresponding locations. Then, the solving method of large-scale linear algebraic equations in six-dimensional form is developed by Cholesky decomposition. As the matrix manipulation is needed in the Galerkin projection, large-scale matrix manipulating method in six-dimensional formulation is studyed to improve the speed of numerical computation.
Third, a direct orthogonal method is proposed to construct a subspace which is used in the Galerkin projection. The method is used in some vehicle design problems to evaluate the computational cost and accuracy. When constructing the subspace, the finite element solutions derived from the sampling points are orthogonalized and normalized by using the single value decomposion technique. The left single vectors are orthogonal and used as the basis of the subspace. The subspace constructed by the direct orthogonal method is computational efficient compared with the subapace derived by the traditional Greedy adaptive method.
Fourth, to further improve the computational efficiency, a hierarchical adaptive method is developed to construct the subspace based on the greedy adaptive method and the direct orthogonal method. The process of constructing the subspace is devided into two phases, which are expressed as the initial subspace and the ultimate subspace. The initial subspace is constructed using the direct orthogonal method and the ultimate subspace is constructed by updating the initial subspace using the greedy adaptive method. The computational accuracy is improved and the computational cost is saved using the hierarchical adaptive method. The effect of the dimension of the initial subspace is analyzed with respect to the computational cost and accuracy.
Finally, an automatic and rapid computational method is proposed to simplifying the computational procedure. The automatic method is developed based on the automatic parameter domain discretization, finite element method, subspace construction and subspace updating techniques. The computational procedure and computational error evaluation of traditional reduced-basis method are changed completely and the offline-online phase is cancelled. It is a new method with real-time subspace construction, real-time subspace updating and real-time error evaluation.
The proposed methods are tested on different vehicle design problems. It is found that the proposed methods are of higher accuracy and lower computational cost compared with the traditional reduced-basis methods. It is proved that the proposed methods are efficient in vehicle body design problems and are applicable to many other design contexts, especially in the real-time situation. For shell structural optimization in engineering practice, the proposed techniques provide an efficient way to achieve an optimal design solution. It is also suitable for other structures with a little change in the decomposition of finite element formulation, such as solid element and beam element.
快速计算方法的基本思路是当系统由多个参数来描述时,这些参数组合会使系统有不同的响应,这些响应可以由有限次试验得到,而系统在新参数下的解可以用事先设计的近似解空间来表示。在实际计算过程中,应用迦辽金映射技术,将结构设计问题向一个事先构造的子空间中进行投影,将大规模问题缩减为小规模问题,该小规模问题即可快速求解。这种方法在对结构进行自由度缩减时能很好地保持原系统的物理属性。论文从以下方面展开工作。
第一,从壳单元的有限元格式入手,对壳单元刚度矩阵在单元水平上进行参数化分解,同时,按照设计问题的具体要求,将具有同一参数特征的单元刚度矩阵按照有限元方法的基本原理进行坐标转换和组装,最终将整体刚度矩阵表达为一种显式参数化格式。显式参数化格式保持了设计问题的参数特性。
第二,提出一种新的刚度矩阵存储技术,将总体刚度矩阵按照节点的物理属性进行存储,大幅节约了存储空间。对单元刚度矩阵的组成和物理意义进行分析,将单元刚度矩阵按照节点大小的顺序排列成六维的矩阵形式；整体坐标系下的刚度矩阵的六维存储格式通过对单元刚度矩阵在对应位置上的组装实现。在六维格式的基础上,对六维格式的刚度矩阵进行Cholesky分解,并求解线性方程组；考虑到在快速计算方法中需要对矩阵进行的投影运算,对六维存储格式的刚度矩阵进行投影运算研究。
第三,提出一种直接正交构造子空间的方法,将该方法应用于某车身结构的快速计算中,评价计算的精度和计算成本。直接正交法在进行子空间构造时,首先使用奇异值分解对样本点的有限元解进行正交归一化,左奇异向量即为所构造的近似子空间的一个基。直接正交法比传统的贪婪算法在计算成本方面具有较大的优势。
第四,为了进一步提高快速计算方法的计算效率,提出一种分级自适应方法。该方法以直接正交法和贪婪算法为基础,将构造子空间的过程分为初级子空间构造过程和最终子空间构造过程。在初级子空间构造过程中使用直接正交法进行计算,以节省计算量为主要目的,在子空间维数较少的情况下,直接正交法的数值损失较小；在初级子空间的基础上使用贪婪算法进行最终子空间的构造,通过误差控制保证所构造子空间的精度。同时,对初始子空间的维数对最终计算的精度和计算量的影响进行了研究。
第五,针对快速计算方法的较为复杂的计算流程,提出一种自动快速计算方法,该方法通过参数域自动离散、有限元计算、子空间构造和子空间实时更新技术实现了快速计算方法的自动化。通过自动快速计算方法的实现,改变了传统的基于减基法基本思想的快速计算方法的基本计算流程和误差评价方法,形成了一种实时构造、实时更新、实时误差评价的新的快速计算方法。
在研究过程中,使用多个车身设计算例对提出的方法进行验证,结果表明,文中提出的快速计算方法在精度和效率上都比常规的减基法有较大的提高。文中提出的方法在车身设计问题中具有较高的计算效率,并可以将其应用于其它的设计工况中,特别适用于有实时计算要求的场合。对于工程实际中的壳结构优化问题,使用文中提出的方法可以快速地得到最优解。同时,通过改变参数化有限元格式,文中提出的方法同样可以用于梁结构以及实体结构等的设计中。
Vehicle body design phase is a time-consuming process which requires the solution of a large-scale problem with many parameters. A new computational method is proposed in the paper to achieve a rapid way of solving the large-scale linear algebraic equations with many parameters in vehicle design process with respect to the computational cost and the computational accuracy.
The basic idea of the method is that although the field variable generally belongs to the infinite-dimensional space associated with the underlying partial differential equation, indeed, it resides on a low-dimensional subspace induced by the parametric dependence. In computational practice, the large-scale system is reduced to a smaller one by Galerkin projection based on that subspace. The structural behaviors are predicted using the reduced system. The physical property of the problem is retained during the projection.The research is carried out in following sections.
First, the finite element formulation of shell element is analyzed and the element stiffness matrix is decomposed to achieve a parametrized formulation on the element level. Then the element matrices with the same parametric properties are assembled based on the finite element method via coordinate transforming. The parametric property is retained in the form of the explicit parametrized finite element formulation.
Second, a new storage method of large-scale matrix named six-dimensional method is proposed to save the computational cost and improve the computational accuracy. In the new storage formulation, the global stiffness matrix is stored according to the node information which indicates the physical properties of each node. The components of the element stiffness matrices are analyzed first and the locations of the element stiffness matrices in six-dimensional matrix are determined according to the order of the nodes. The global stiffness matrix in six-dimensional form is achieved by assembling the element stiffness in the corresponding locations. Then, the solving method of large-scale linear algebraic equations in six-dimensional form is developed by Cholesky decomposition. As the matrix manipulation is needed in the Galerkin projection, large-scale matrix manipulating method in six-dimensional formulation is studyed to improve the speed of numerical computation.
Third, a direct orthogonal method is proposed to construct a subspace which is used in the Galerkin projection. The method is used in some vehicle design problems to evaluate the computational cost and accuracy. When constructing the subspace, the finite element solutions derived from the sampling points are orthogonalized and normalized by using the single value decomposion technique. The left single vectors are orthogonal and used as the basis of the subspace. The subspace constructed by the direct orthogonal method is computational efficient compared with the subapace derived by the traditional Greedy adaptive method.
Fourth, to further improve the computational efficiency, a hierarchical adaptive method is developed to construct the subspace based on the greedy adaptive method and the direct orthogonal method. The process of constructing the subspace is devided into two phases, which are expressed as the initial subspace and the ultimate subspace. The initial subspace is constructed using the direct orthogonal method and the ultimate subspace is constructed by updating the initial subspace using the greedy adaptive method. The computational accuracy is improved and the computational cost is saved using the hierarchical adaptive method. The effect of the dimension of the initial subspace is analyzed with respect to the computational cost and accuracy.
Finally, an automatic and rapid computational method is proposed to simplifying the computational procedure. The automatic method is developed based on the automatic parameter domain discretization, finite element method, subspace construction and subspace updating techniques. The computational procedure and computational error evaluation of traditional reduced-basis method are changed completely and the offline-online phase is cancelled. It is a new method with real-time subspace construction, real-time subspace updating and real-time error evaluation.
The proposed methods are tested on different vehicle design problems. It is found that the proposed methods are of higher accuracy and lower computational cost compared with the traditional reduced-basis methods. It is proved that the proposed methods are efficient in vehicle body design problems and are applicable to many other design contexts, especially in the real-time situation. For shell structural optimization in engineering practice, the proposed techniques provide an efficient way to achieve an optimal design solution. It is also suitable for other structures with a little change in the decomposition of finite element formulation, such as solid element and beam element.
引文
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