基于改进位移模式的有限元超收敛算法研究
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摘要
超收敛计算的研究是近年来有限元领域研究中的热点与难点之一。通过对基于改进位移模式的积分形式与基于常规位移模式伽辽金方程的比较，导出了伽辽金方程精确成立的条件。在以此条件作为位移模式分析的理论依据前提下，本文提出了改进位移模式，即将高阶有限元解的位移模式用常规有限元解的位移模式表示，用两种位移模式之和构造新的位移模式，结合多尺度方法的思想和超收敛计算的解析公式，提出了一种全新的前处理超收敛的计算方法。该方法无需任何人为的磨光，对一维有限元法的线性单元，本文结点和单元的位移、导数都达到了h4阶的超收敛精度。应力跨单元自动平衡，可精确满足自然边界条件，且实施简便，计算量增加很小。
有限元线法是一个不断发展的课题，国内外对有限元线法超收敛的计算研究得较少。本文利用有限元线法的半解析性，将一维有限元中获得全面成功的改进位移模式的超收敛算法推广至二维有限元线法，取得了良好的效果。
全文主要工作如下：
1、通过基于改进位移模式与基于常规位移模式伽辽金方程的比较，导出了伽辽金方程精确成立的条件。论证了位移模式与泡函数的相关性。提出了一次元与泡函数结合的单元方案。利用精确成立的条件，通过量级分析，保留泡函数主要的影响项，避免了求解泡函数的解析表达式。为位移模式的分析提供了理论依据。
2、以一维C0问题为模型问题，提出有限元法的基于改进位移模式的前处理超收敛算法。在原有试函数的基础上，增加了高阶试函数，使得单元内平衡方程的残差减少，从而达到提高精度的目标。对于近似单元，根据单元内部平衡条件，导出单元上任一点的位移和导数的超收敛解的计算公式。基于伽辽金方法，采用积分形式推导了单元刚度矩阵。
3、将基于改进位移模式的前处理超收敛算法成功推广到一维有限元法其它问题，包括(1)一维C1问题；(2)二阶非自伴两点边值问题Galerkin有限元问题，亦即将基于改进位移模式的前处理超收敛算法推广到了非自伴算子问题而不仅仅是自伴算子的问题。（3）一维n阶问题的超收敛算法，本部分工作为该法广泛应用于一般一维问题的有限元法的超收敛计算打下了良好的基础。
4、将基于改进位移模式的前处理超收敛算法成功推广应用到二维有限元线法的Poisson方程问题，基于线性形函数，采用变分形式推导了有限元线法求解的修正的常微分方程组。算例结果表明：结点和单元内的位移、导数的收敛精度得到了极大的提高。
由于本文得出的应力和位移是逐点超收敛的，因而有望在此基础上发展出不同于目前常规的误差估计和自适应求解方法。
Research on super-convergent computation in the field of Finite ElementMethod(FEM) has been a hot and difficult subject in recent years. Based on thecomparison of improved displacement mode and conventional displacement mode ofGalerkin equations, the exact conditions for the establishment of Galerkin equation isderived. Based on the premise of this paper, improved displacement mode isproposed,the displacement mode of high-order finite element solution is expressedwith a conventional finite element solution, the new displacement model isconstructed with the sum of the displacement model of conventional finite elementand the high-order displacement model of finite element solution, combined with thethinking of multi-scale method and the analytical formulas of superconvergencecomputing, a new pretreatment method of superconvergence calculation is proposed.The method does not need any artificial polish, for one-dimensional FEM, thedisplacement and the derivative accuracy of nodes and elements have reached h4order. The stresses so calculated are automatically in equilibrium across elements andare exact at free ends or edges, the implementation is simple, the work of calculationincrease very small.
The Finite Element Method of Lines(FEMOL) is a continuous developmentsubject,At home and abroad study of superconvergence calculation on FEMOL islittle.Used the semi-analytical nature of FEMOL,the superconvergence calculation ofimproved displacement mode is also applied, to two-dimensional FEMOL and similarconclusions have been made.
The main work of this paper is as follows:
1. Based on the comparison of improved displacement mode and conventionaldisplacement mode of Galerkin equations, the exact conditions for the establishmentof Galerkin equation is derived. The association of displacement mode and the bubblefunction is demonstrated. The combination program of liner element and bubblefunction is proposed. Using the condition of precise establishment, by the analysisorder of magnitude, retaining the main impact item of bubble function, the analyticalexpression solving of bubble function is avoided. The theoretical basis of the analysisof the displacement mode is provided.
2. Taking one-dimensional C0problem as a model problem, a pretreatment method of superconvergence calculation based on improved displacement mode isproposed for FEM. for improving the accuracy of the solution and reducing theresidual of balance equation, the higher-order trial functions is added at thefoundation of the original trial function, for approximate elements, according to theinternal equilibrium of unit,a complete set of formulas for super-convergentdisplacements and derivatives at any point on an element are also derived. Theelement stiffness matrix is derived using the integral form.
3. A pretreatment method of superconvergence calculation based on improveddisplacement mode is successfully extended to other problems in one-dimensionalFEM,such as one-dimensional C1problem and the for second order non-self-adjointboundary-value problem, now that, the method can be used for not only self-adjointoperator problem but also non-self-adjoint operator problem. The problem ofone-dimensional n-order can be sloved by this method.
4. A pretreatment method of superconvergence calculation based on improveddisplacement mode is successfully extended to Poisson equation problems intwo-dimensional FEMOL, based on linear shape function the modified ordinarydifferential equations of FEMOL is derived using the variational form. The result ofexamples show that the displacement and the derivative accuracy of nodes andelements has been greatly improved.
Using super-convergent stresses and displacements, the associated errorestimation and self-adaptive solution strategy may also be developed in the future.
有限元线法是一个不断发展的课题，国内外对有限元线法超收敛的计算研究得较少。本文利用有限元线法的半解析性，将一维有限元中获得全面成功的改进位移模式的超收敛算法推广至二维有限元线法，取得了良好的效果。
全文主要工作如下：
1、通过基于改进位移模式与基于常规位移模式伽辽金方程的比较，导出了伽辽金方程精确成立的条件。论证了位移模式与泡函数的相关性。提出了一次元与泡函数结合的单元方案。利用精确成立的条件，通过量级分析，保留泡函数主要的影响项，避免了求解泡函数的解析表达式。为位移模式的分析提供了理论依据。
2、以一维C0问题为模型问题，提出有限元法的基于改进位移模式的前处理超收敛算法。在原有试函数的基础上，增加了高阶试函数，使得单元内平衡方程的残差减少，从而达到提高精度的目标。对于近似单元，根据单元内部平衡条件，导出单元上任一点的位移和导数的超收敛解的计算公式。基于伽辽金方法，采用积分形式推导了单元刚度矩阵。
3、将基于改进位移模式的前处理超收敛算法成功推广到一维有限元法其它问题，包括(1)一维C1问题；(2)二阶非自伴两点边值问题Galerkin有限元问题，亦即将基于改进位移模式的前处理超收敛算法推广到了非自伴算子问题而不仅仅是自伴算子的问题。（3）一维n阶问题的超收敛算法，本部分工作为该法广泛应用于一般一维问题的有限元法的超收敛计算打下了良好的基础。
4、将基于改进位移模式的前处理超收敛算法成功推广应用到二维有限元线法的Poisson方程问题，基于线性形函数，采用变分形式推导了有限元线法求解的修正的常微分方程组。算例结果表明：结点和单元内的位移、导数的收敛精度得到了极大的提高。
由于本文得出的应力和位移是逐点超收敛的，因而有望在此基础上发展出不同于目前常规的误差估计和自适应求解方法。
Research on super-convergent computation in the field of Finite ElementMethod(FEM) has been a hot and difficult subject in recent years. Based on thecomparison of improved displacement mode and conventional displacement mode ofGalerkin equations, the exact conditions for the establishment of Galerkin equation isderived. Based on the premise of this paper, improved displacement mode isproposed,the displacement mode of high-order finite element solution is expressedwith a conventional finite element solution, the new displacement model isconstructed with the sum of the displacement model of conventional finite elementand the high-order displacement model of finite element solution, combined with thethinking of multi-scale method and the analytical formulas of superconvergencecomputing, a new pretreatment method of superconvergence calculation is proposed.The method does not need any artificial polish, for one-dimensional FEM, thedisplacement and the derivative accuracy of nodes and elements have reached h4order. The stresses so calculated are automatically in equilibrium across elements andare exact at free ends or edges, the implementation is simple, the work of calculationincrease very small.
The Finite Element Method of Lines(FEMOL) is a continuous developmentsubject,At home and abroad study of superconvergence calculation on FEMOL islittle.Used the semi-analytical nature of FEMOL,the superconvergence calculation ofimproved displacement mode is also applied, to two-dimensional FEMOL and similarconclusions have been made.
The main work of this paper is as follows:
1. Based on the comparison of improved displacement mode and conventionaldisplacement mode of Galerkin equations, the exact conditions for the establishmentof Galerkin equation is derived. The association of displacement mode and the bubblefunction is demonstrated. The combination program of liner element and bubblefunction is proposed. Using the condition of precise establishment, by the analysisorder of magnitude, retaining the main impact item of bubble function, the analyticalexpression solving of bubble function is avoided. The theoretical basis of the analysisof the displacement mode is provided.
2. Taking one-dimensional C0problem as a model problem, a pretreatment method of superconvergence calculation based on improved displacement mode isproposed for FEM. for improving the accuracy of the solution and reducing theresidual of balance equation, the higher-order trial functions is added at thefoundation of the original trial function, for approximate elements, according to theinternal equilibrium of unit,a complete set of formulas for super-convergentdisplacements and derivatives at any point on an element are also derived. Theelement stiffness matrix is derived using the integral form.
3. A pretreatment method of superconvergence calculation based on improveddisplacement mode is successfully extended to other problems in one-dimensionalFEM,such as one-dimensional C1problem and the for second order non-self-adjointboundary-value problem, now that, the method can be used for not only self-adjointoperator problem but also non-self-adjoint operator problem. The problem ofone-dimensional n-order can be sloved by this method.
4. A pretreatment method of superconvergence calculation based on improveddisplacement mode is successfully extended to Poisson equation problems intwo-dimensional FEMOL, based on linear shape function the modified ordinarydifferential equations of FEMOL is derived using the variational form. The result ofexamples show that the displacement and the derivative accuracy of nodes andelements has been greatly improved.
Using super-convergent stresses and displacements, the associated errorestimation and self-adaptive solution strategy may also be developed in the future.
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