各向异性及双参数非协调有限元方法研究
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摘要
近年来，各向异性有限元方法已成为有限元领域的热点问题，陆续出现了许多有关此方面的理论及应用研究成果，见[26，27，53，54]，其中大部分工作主要是针对二阶或四阶椭圆边值问题协调与非协调元的插值误差估计进行的。目前，各向异性有限元方法的主要挑战性工作有：
(1)．由于在各向异性网格剖分下，Bramble-Hilbert引理[4]不能直接引用，因此插值误差的估计无法按照传统的有限元误差估计技巧进行。对于非协调有限元来说，相容误差的估计是十分困难的，因为当网格剖分不再满足正则假设，拟一致假设或反假设时，相容误差中有关边界项的估计，将出现的因子|F|／|K|，当F是单元K的最长边时，该因子可能会趋于无穷大，无法保证收敛性，需要探索新的途径和边界估计技巧。
(2)．已有的一些单元是不具备各向异性特征[77]的，如Th.Apel在[53]中已证明，较新的旋转Q_1元[36]是不能用于各向异性网格的，并且举出了反例；
(3)．对于混合元方法来说，插值算子在各向异性网格下的适定性，稳定性以及LBB条件(混合有限元方法的关键)等验证工作都有和大的难度。
另外，超收敛研究也是有限元领域备受关注的问题之一，出现了许多高精度有限元的研究，但是，基本上也是在正则网格剖分下讨论的，有关各向异性单元的超收敛性质的研究则相对较少，且难度也较大，主要是因为后验插值算子的构造以及各向异性特征的验证等都十分困难。
另一方面，如何构造高效的板单元的研究一直是有限元领域的热点和难点，由于”双参数”有限元具有自由度小并能兼顾收敛性的特点，本文利用双参数有限元的思想构造了两个可用于四阶板问题的单元(一个是具有各向异性特征的12参矩形元，另一个是三角形九参数元)
本文主要系统研究两大类问题：一是选取几个典型的各向异性非协调元，如矩形Wilson元，三角形Carey元，一类Crouzeix-Raviart型元，五节点元等，分别对位移障碍下变分不等式问题，二阶椭圆边值问题，Stokes问题和平面弹性问题做了进一步深入探讨；二是并利用双参数有限元法构造了8自由度12参矩形板元和三角形九参元，对矩形板元我们进行了超收敛分析，对双参数三角形板元分析了其收敛性并给出了数值实验，结果表明理论分析和实际计算是相吻合的。
具体地讲，位移障碍下变分不等式问题的各向异性非协调有限元方法主要由两部分组成：
(1)以两个著名的非协调有限元矩形Wilson元[20]和三角形Carey元[9]为例，研究位移障碍下变分不等式问题的各向异性非协调有限元方法，通过引入新的技巧，得到了与传统有限元方法相同的最优误差估计，这一部分中使用的方法对一类可分离出协调部分和非协调部分，且通过Irons分片检查的单元均成立；
(2)讨论了各向异性网格下位移障碍下变分不等式问题的一类Crouzeix-Raviart型非协调有限元(特别是三角形元)逼近，通过另外一种完全不同于(1)的新的误差估计技巧，在自由边界长度有限及精确解的正则性假设条件下，得到了与[1，11，51]相同的最优误差估计。本文的结果再次证实了一个重要结论，即对于位移障碍下变分不等式的非协调有限元方法的收敛性分析，网格剖分的正则性假设不是必要的，从而拓宽了有限元尤其是非协调有限元的应用范围。
其次本文又利用三角形Carey元研究了二阶椭圆问题的收敛性及误差估计。在已有的相关文献中，大多数要求问题的解u∈H~2(Ω)或u∈H~3(Ω)，[4]、[52]和[50]针对u∈H~4(Ω)时研究了协调线性三角形元的收敛性。以[4]为基础，[49]和[52]在同样条件下对非协调有限元做了探讨。上述文献均要求网格剖分满足正则假设或拟一致假设。本文利用不同于文献[49，50，52]的新的技巧和方法，在各向异性网格剖分和解的弱正则条件下，得到了和他们同样的结果。并且，本文中的结果对其他的一些非协调有限元，如各向异性Wilson元，各向异性类Wilson元等也成立。
再次，本文研究二阶椭圆问题和Stokes问题的低阶元混合有限元方法。对二阶问题，构造了一个新的单元格式，当其精确解具有低正则性时，在不要求网格满足正则假设或拟一致假设的条件下，我们得到了最优误差估计，与[33]相比，我们的证明方法较为简单；对于Stokes问题，研究五节点非协调矩形元对它的逼近，通过引入全新的估计方法，给出了关于速度、压力的超逼近性质。同时，通过巧妙的构造了一个适当的插值后处理算子，技巧性的导出了在各向异性网格下的超收敛结果，丰富了有限元(特别是非协调有限元)方法的内容。本文的结果同样适用于[36，48]在正则网格下所讨论的的旋转Q_1元。
接下来，本文研究了平面弹性问题，结合[29，78]的思想，直接构造了一个新的可以用于各向异性网格的矩形非协调元，利用其所具有的特殊性质，并通过引入辅助空间及新颖的估计方法，在各向异性网格剖分下对纯位移平面弹性问题的Locking现象进行了研究，发现该元是Locking-Free的，同时得到了最优能量范数和L~2范数误差估计。值得指出的是由于这里的估计技巧非常特殊(特别是关于相容误差的估计)，完全不同于以往正则假设下的非协调元误差分析，因而更具有理论意义和应用价值，从而拓宽了各向异性有限元方法(尤其是非协调元)的应用范围。
最后，利用双参数有限元法构造了8自由度12参矩形板元和三角形九参元：
(1)通过[47]中的单元上的形函数空间修正为P_2∪span{x~3，y~3}，节点参数取为顶点函数值与一阶偏导值，构造了一个新的8自由度12参矩形板元，利用单元的特殊构造和全新的思路和技巧，证明了在各向异性条件下证明了该单元的收敛阶也可以达到O(h~2)阶。同时，利用Bramble-Hilbert引理证明了该元有O(h~2)阶的超收敛性。上述结论在目前相关的文献中还未见报道；
(2)通过对Morley元的改进，构造了一个新的九参数三角形板元。新单元的形函数空间和实际节点参数分别同Morley元及Zienkiewicz元相同。因此，新单元的总体自由度只有Morley元的3／4。理论和数值实验结果表明，新单元形函数的外法向导数在单元间连续，通过F-E-M-Test，具有计算方便、收敛效果更好的优点，而且自由度选取对称，对任意三角形剖分收敛，克服了Zienkiewicz元和广义协调元的缺陷。同时，新单元对弯距的计算效果比Morley元和Zienkiewicz元都要好。不仅如此，本文最后对新单元的数值结果进行了外推运算，其数值效果得到了很大的改善。因此，该元是目前为止较为理想的有应用价值的板单元。
Recently, a lot of studies have been focused on the anisotropic finite element method, and there appears many relative papers(see [26,27,53,54]). However, the main attention of those were paid to the study of conforming Lagrange elements and the interpolation error analysis on the second order or fourth order elliptic boundary value problems. At present, the challenge of anisotropic finite element method is as follows:
(1). Since the Bramble-Hilbert Lemma [4] can not be used directly on the anisotropic meshes, so the interpolation error can not be estimated by the use of the conventional finite element methods. Furthermore, the consistency error estimate is the key of the nonconforming finite element analysis, and it will become very difficult to be dealt with because there will appear a factor |F|/|K| which tends to∞when the estimate is made on the longer sides F of the element K.
(2). Some elements didn't have anisotropy [77], for example, in [53], Th.Apel proved that the rotated Q_1 element [36] can not be used on anisotropic meshes by giving a counter example.
(3). The proof of the well-posed property, stability and the LBB condition are very difficult to deal with.
In addition, the studies of the superconvergence analysis on anisotropic meshes is another open problem. It is also very hard to deal with since the construction of the post interpolation operator and the proof of the anisotropic property are challengeable.
In this paper, we focus on the the study of anisotropic nonconforming element of the variational inequality problem with displacement obstacle, the second order elliptic problem, Stokes problem, and the planar elasticity problem. And rectangle and triangle plate elements are constructed respectively by taking advantage of the double set parameter methods. For the first element, the superconvergence result are proposed; and for the second one, we discuss its convergence and the numerical results are given to demonstrate the validity of our theoretical analysis. At the last part of the paper, the construction of two double set parameter elements and their convergence are discussed.
In detail, the arrangement of the paper is as follows:
Firstly, we first study the nonconforming finite element (Wilson element [20] and Carey element [9]) approximation to the second order variational obstacle problem with displacement obstacle on the anisotropic meshes. By using novel approaches, the convergence analysis is given and the optimal error estimates are obtained. The method proposed herein is also valid to the other element which can be separated into conforming part and nonconforming parts and pass the Irons patch test. Next, a class of Crouzeix-Raviart type nonconforming finite element approximations are considered for solving the above problem on anisotropic meshes. The approaches for this kind of element is totally different from that of the former, and the convergence analysis is given and the optimal error estimates are obtained under the hypothesis of the finite length of the free boundary.
Secondly, the convergence behavior of a Carey triangle nonconforming element for the second order elliptic problem with lowest regularity on anisotropic meshes are discussed. There have been a lot of studies this aspect when the exact solution u∈H~2(Ω) or u∈H~3(Ω). [4], [52] and [50]studied the convergence of conforming linear triangle elements with minimal regularity assumptions u∈H~4(Ω,), here u is the solutions of the second order elliptic problems. However, their is another vital deficiency of the previous studies, i.e., they relies on the regularity assumption or quasi-uniform assumption on the subdivision of meshes. By using different techniques from that of [49, 50, 52], the same results as those of [49,50,52] are obtained on the anisotropic meshes with the minimal regularity assumption.
Thirdly, we discuss the the second order elliptic problems and Stokes problem. For the first one, a new element is constructed and the optimal error estimates are obtained under the lowest regularity of the exact solution. In addition, the proof of this paper is much easier in comparison with [33]. For the second one, with the use of five nodal rectangular element, the superclose results of the velocity and the pressure are obtained by novel techniques, and furthermore, by constructing a proper interpolation post operator, the superconvergence can be deduced on the anisotropic meshes. The main result of the second one is also valid to the rotated Q_1 element in [36,48] on regular meshes.
Next, we study the the planar elasticity problem with pure displacement boundary condition with the use of a new anisotropic nonconforming rectangular finite element which is constructed in the paper. It is proved that this element is locking-free when the Laméconstantλ→∞. By use of the special properties of this element, and by introducing the complementary space and a series of novel techniques, the optimal error estimates of the energy norm and the L~2-norm are obtained. Thus we get rid of the restriction of regularity assumption, quasi-uniform assumption or the inverse assumptions on the meshes required in the conventional finite element methods analysis, and extend the applicable scope of the nonconforming finite elements.
Finally, we construct 12 parameters rectangular plate element(by modifying the shape function space of [47] as P_3∪span{x~3,y~3}) and 9 parameter triangle plate ele-ment(by modifying Morley element) by use of the double set parameter finite element methods. For the first element, we can prove that the consistency error is O(h~2) order which can be obtained on anisotropic meshes. For the second one, its shape function space and the real node parameter is as same as those of Morley element and Zienkiewicz element, but the total degrees of freedoms is only three quarters of that of Morley element. The theoretical analysis and the numerical test show that the element is very excellent for the reason that the outer normal derivative is continuous, degrees of freedom is symmetrical, and it is convergent for arbitrary triangle subdivision.
(1)．由于在各向异性网格剖分下，Bramble-Hilbert引理[4]不能直接引用，因此插值误差的估计无法按照传统的有限元误差估计技巧进行。对于非协调有限元来说，相容误差的估计是十分困难的，因为当网格剖分不再满足正则假设，拟一致假设或反假设时，相容误差中有关边界项的估计，将出现的因子|F|／|K|，当F是单元K的最长边时，该因子可能会趋于无穷大，无法保证收敛性，需要探索新的途径和边界估计技巧。
(2)．已有的一些单元是不具备各向异性特征[77]的，如Th.Apel在[53]中已证明，较新的旋转Q_1元[36]是不能用于各向异性网格的，并且举出了反例；
(3)．对于混合元方法来说，插值算子在各向异性网格下的适定性，稳定性以及LBB条件(混合有限元方法的关键)等验证工作都有和大的难度。
另外，超收敛研究也是有限元领域备受关注的问题之一，出现了许多高精度有限元的研究，但是，基本上也是在正则网格剖分下讨论的，有关各向异性单元的超收敛性质的研究则相对较少，且难度也较大，主要是因为后验插值算子的构造以及各向异性特征的验证等都十分困难。
另一方面，如何构造高效的板单元的研究一直是有限元领域的热点和难点，由于”双参数”有限元具有自由度小并能兼顾收敛性的特点，本文利用双参数有限元的思想构造了两个可用于四阶板问题的单元(一个是具有各向异性特征的12参矩形元，另一个是三角形九参数元)
本文主要系统研究两大类问题：一是选取几个典型的各向异性非协调元，如矩形Wilson元，三角形Carey元，一类Crouzeix-Raviart型元，五节点元等，分别对位移障碍下变分不等式问题，二阶椭圆边值问题，Stokes问题和平面弹性问题做了进一步深入探讨；二是并利用双参数有限元法构造了8自由度12参矩形板元和三角形九参元，对矩形板元我们进行了超收敛分析，对双参数三角形板元分析了其收敛性并给出了数值实验，结果表明理论分析和实际计算是相吻合的。
具体地讲，位移障碍下变分不等式问题的各向异性非协调有限元方法主要由两部分组成：
(1)以两个著名的非协调有限元矩形Wilson元[20]和三角形Carey元[9]为例，研究位移障碍下变分不等式问题的各向异性非协调有限元方法，通过引入新的技巧，得到了与传统有限元方法相同的最优误差估计，这一部分中使用的方法对一类可分离出协调部分和非协调部分，且通过Irons分片检查的单元均成立；
(2)讨论了各向异性网格下位移障碍下变分不等式问题的一类Crouzeix-Raviart型非协调有限元(特别是三角形元)逼近，通过另外一种完全不同于(1)的新的误差估计技巧，在自由边界长度有限及精确解的正则性假设条件下，得到了与[1，11，51]相同的最优误差估计。本文的结果再次证实了一个重要结论，即对于位移障碍下变分不等式的非协调有限元方法的收敛性分析，网格剖分的正则性假设不是必要的，从而拓宽了有限元尤其是非协调有限元的应用范围。
其次本文又利用三角形Carey元研究了二阶椭圆问题的收敛性及误差估计。在已有的相关文献中，大多数要求问题的解u∈H~2(Ω)或u∈H~3(Ω)，[4]、[52]和[50]针对u∈H~4(Ω)时研究了协调线性三角形元的收敛性。以[4]为基础，[49]和[52]在同样条件下对非协调有限元做了探讨。上述文献均要求网格剖分满足正则假设或拟一致假设。本文利用不同于文献[49，50，52]的新的技巧和方法，在各向异性网格剖分和解的弱正则条件下，得到了和他们同样的结果。并且，本文中的结果对其他的一些非协调有限元，如各向异性Wilson元，各向异性类Wilson元等也成立。
再次，本文研究二阶椭圆问题和Stokes问题的低阶元混合有限元方法。对二阶问题，构造了一个新的单元格式，当其精确解具有低正则性时，在不要求网格满足正则假设或拟一致假设的条件下，我们得到了最优误差估计，与[33]相比，我们的证明方法较为简单；对于Stokes问题，研究五节点非协调矩形元对它的逼近，通过引入全新的估计方法，给出了关于速度、压力的超逼近性质。同时，通过巧妙的构造了一个适当的插值后处理算子，技巧性的导出了在各向异性网格下的超收敛结果，丰富了有限元(特别是非协调有限元)方法的内容。本文的结果同样适用于[36，48]在正则网格下所讨论的的旋转Q_1元。
接下来，本文研究了平面弹性问题，结合[29，78]的思想，直接构造了一个新的可以用于各向异性网格的矩形非协调元，利用其所具有的特殊性质，并通过引入辅助空间及新颖的估计方法，在各向异性网格剖分下对纯位移平面弹性问题的Locking现象进行了研究，发现该元是Locking-Free的，同时得到了最优能量范数和L~2范数误差估计。值得指出的是由于这里的估计技巧非常特殊(特别是关于相容误差的估计)，完全不同于以往正则假设下的非协调元误差分析，因而更具有理论意义和应用价值，从而拓宽了各向异性有限元方法(尤其是非协调元)的应用范围。
最后，利用双参数有限元法构造了8自由度12参矩形板元和三角形九参元：
(1)通过[47]中的单元上的形函数空间修正为P_2∪span{x~3，y~3}，节点参数取为顶点函数值与一阶偏导值，构造了一个新的8自由度12参矩形板元，利用单元的特殊构造和全新的思路和技巧，证明了在各向异性条件下证明了该单元的收敛阶也可以达到O(h~2)阶。同时，利用Bramble-Hilbert引理证明了该元有O(h~2)阶的超收敛性。上述结论在目前相关的文献中还未见报道；
(2)通过对Morley元的改进，构造了一个新的九参数三角形板元。新单元的形函数空间和实际节点参数分别同Morley元及Zienkiewicz元相同。因此，新单元的总体自由度只有Morley元的3／4。理论和数值实验结果表明，新单元形函数的外法向导数在单元间连续，通过F-E-M-Test，具有计算方便、收敛效果更好的优点，而且自由度选取对称，对任意三角形剖分收敛，克服了Zienkiewicz元和广义协调元的缺陷。同时，新单元对弯距的计算效果比Morley元和Zienkiewicz元都要好。不仅如此，本文最后对新单元的数值结果进行了外推运算，其数值效果得到了很大的改善。因此，该元是目前为止较为理想的有应用价值的板单元。
Recently, a lot of studies have been focused on the anisotropic finite element method, and there appears many relative papers(see [26,27,53,54]). However, the main attention of those were paid to the study of conforming Lagrange elements and the interpolation error analysis on the second order or fourth order elliptic boundary value problems. At present, the challenge of anisotropic finite element method is as follows:
(1). Since the Bramble-Hilbert Lemma [4] can not be used directly on the anisotropic meshes, so the interpolation error can not be estimated by the use of the conventional finite element methods. Furthermore, the consistency error estimate is the key of the nonconforming finite element analysis, and it will become very difficult to be dealt with because there will appear a factor |F|/|K| which tends to∞when the estimate is made on the longer sides F of the element K.
(2). Some elements didn't have anisotropy [77], for example, in [53], Th.Apel proved that the rotated Q_1 element [36] can not be used on anisotropic meshes by giving a counter example.
(3). The proof of the well-posed property, stability and the LBB condition are very difficult to deal with.
In addition, the studies of the superconvergence analysis on anisotropic meshes is another open problem. It is also very hard to deal with since the construction of the post interpolation operator and the proof of the anisotropic property are challengeable.
In this paper, we focus on the the study of anisotropic nonconforming element of the variational inequality problem with displacement obstacle, the second order elliptic problem, Stokes problem, and the planar elasticity problem. And rectangle and triangle plate elements are constructed respectively by taking advantage of the double set parameter methods. For the first element, the superconvergence result are proposed; and for the second one, we discuss its convergence and the numerical results are given to demonstrate the validity of our theoretical analysis. At the last part of the paper, the construction of two double set parameter elements and their convergence are discussed.
In detail, the arrangement of the paper is as follows:
Firstly, we first study the nonconforming finite element (Wilson element [20] and Carey element [9]) approximation to the second order variational obstacle problem with displacement obstacle on the anisotropic meshes. By using novel approaches, the convergence analysis is given and the optimal error estimates are obtained. The method proposed herein is also valid to the other element which can be separated into conforming part and nonconforming parts and pass the Irons patch test. Next, a class of Crouzeix-Raviart type nonconforming finite element approximations are considered for solving the above problem on anisotropic meshes. The approaches for this kind of element is totally different from that of the former, and the convergence analysis is given and the optimal error estimates are obtained under the hypothesis of the finite length of the free boundary.
Secondly, the convergence behavior of a Carey triangle nonconforming element for the second order elliptic problem with lowest regularity on anisotropic meshes are discussed. There have been a lot of studies this aspect when the exact solution u∈H~2(Ω) or u∈H~3(Ω). [4], [52] and [50]studied the convergence of conforming linear triangle elements with minimal regularity assumptions u∈H~4(Ω,), here u is the solutions of the second order elliptic problems. However, their is another vital deficiency of the previous studies, i.e., they relies on the regularity assumption or quasi-uniform assumption on the subdivision of meshes. By using different techniques from that of [49, 50, 52], the same results as those of [49,50,52] are obtained on the anisotropic meshes with the minimal regularity assumption.
Thirdly, we discuss the the second order elliptic problems and Stokes problem. For the first one, a new element is constructed and the optimal error estimates are obtained under the lowest regularity of the exact solution. In addition, the proof of this paper is much easier in comparison with [33]. For the second one, with the use of five nodal rectangular element, the superclose results of the velocity and the pressure are obtained by novel techniques, and furthermore, by constructing a proper interpolation post operator, the superconvergence can be deduced on the anisotropic meshes. The main result of the second one is also valid to the rotated Q_1 element in [36,48] on regular meshes.
Next, we study the the planar elasticity problem with pure displacement boundary condition with the use of a new anisotropic nonconforming rectangular finite element which is constructed in the paper. It is proved that this element is locking-free when the Laméconstantλ→∞. By use of the special properties of this element, and by introducing the complementary space and a series of novel techniques, the optimal error estimates of the energy norm and the L~2-norm are obtained. Thus we get rid of the restriction of regularity assumption, quasi-uniform assumption or the inverse assumptions on the meshes required in the conventional finite element methods analysis, and extend the applicable scope of the nonconforming finite elements.
Finally, we construct 12 parameters rectangular plate element(by modifying the shape function space of [47] as P_3∪span{x~3,y~3}) and 9 parameter triangle plate ele-ment(by modifying Morley element) by use of the double set parameter finite element methods. For the first element, we can prove that the consistency error is O(h~2) order which can be obtained on anisotropic meshes. For the second one, its shape function space and the real node parameter is as same as those of Morley element and Zienkiewicz element, but the total degrees of freedoms is only three quarters of that of Morley element. The theoretical analysis and the numerical test show that the element is very excellent for the reason that the outer normal derivative is continuous, degrees of freedom is symmetrical, and it is convergent for arbitrary triangle subdivision.
引文
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[18] 陈万吉，唐立民，等参拟协调元，大连工学院学报，1981，20(1)：63-74．
[19] 石钟慈，陈绍春，Specht九参数板元的分析，计算数学，1989，11(3)：312-318．
[20] Z. C. Shi. Convergence properties of two nonconforming finite elements. Comp. Math. Appl. Mech. Engrg., 1985, 48:123-137.
[21] 张鸿庆，王鸣，多套函数有限元逼近及拟协调板元，应用数学与力学，1985，6(1)：41-52．
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[29] Th.Apel, Anisotropic finite elements:local estimate and applications. B.G.Teubner Stuttgart. Leipzig, 1999.
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[31] X.J.Xu, A V-cycle mutigrid method for rotated Q_1 nonconforming element. J.of Comp. Math., 1999, 21(2):251-256.
[32] L.H.Wang, A kind of nonconforming finite element approximations to the fourth order variational inequality problem with displacement obstacle. Chineses J.Comp.Math., 1990, 12(4):352-356.
[33] R. Hiptemair, Nonconforming mixed discretization of second order elliptic problems, Technical University Munchen. Mathematics Institute, M 9404(1994).
[34] M. Farhloul, M. Fortin, A nonconforming mixed finite element for second order elliptic problems, Numer. Mehtods P.D.Eq., 1997, 13: 445-457.
[35] S. C. CHEN, Z.C. SHI, Double set parameter method of constructing a stiffness matrix. Chinese J. Numer. Math, and Appl., 1991, 13(3): 55-69.
[36] R.Rannacher, Turek S, A simple nonconforming quarilateral Stokes element, Numer. Meth. for PDEs, 1992, 8: 97-111.
[37] G.Acosta, R.G.Duran, The maximum angle condition for mixed and nonconforming elements, Application to the Stokes equations, SIAM J.Anal., 1999, 37: 18-36.
[38] R.Becker, R.Rannacher, Finite element solution of the incompressible Navier-Stokes equations on anisotrpoically refined meshes, Fast Solvers for Flow Problems Notes on Numerical Fluid Mechanics wiesbaden: Vieweg, 1995, 49: 52-62.
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[40] H. E1 Bouzid, S. Nicaise, Refined mixed finite element method for the Stokes problem, Analysis, Numerics and Applications of Differential and Integral Equations(Pitman Research Notes in Mathematics, Harlow: Longman, 1998, 379: 158-162.
[41] X. M. BU, Some extensions of double set parameter method and application to rectangular plate element. Numer. Math. Sinica, 1995, 1:65-72.
[42] 石钟慈，陈绍春，九参数拟协调板元的直接分析，计算数学，1990，12(1)：76-84．
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