基于Freefem++的有限元数值计算在计算物理中的应用
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摘要
有限元软件Freefem++以语句简洁,容易上手而著称,非常适合于教学和科研。该软件仅提供直角坐标系的梯度和法方向程序单元,没有极坐标系的,不利用极坐标系物理问题的处理。针对该问题,文章导出了直角坐标系与极坐标系中梯度与法方向的转换关系,成功地应用于1/4圆域的泊松方程中,并设计了基于Freefem++的有限元程序。结果表明,除坐标原点外,数值计算结果与解析解表现出良好的一致性和稳定性,验证了文章中推导的极坐标系梯度和法方向公式的合理性。
The programing sentence of Freefem++is characterized as concise,making the finite element software easy to be used for teaching and scientific research.Its disadvantage also exists for its no supplying gradient and normal programing elements in polar coordinate system,but merely Cartesian system.This drawback makes the software impossible to deal with physical problems in polar coordinate system.In order to tackle the problem,we derived the relation of gradient and normal direction between the two coordinate systems.We applied the transformation to Poisson equation in 1/4 circle domain,and scripted the program in the format of Freefem++.Results indicate that the numerical result of estimated function is consistent with the analytical solution,except those around the coordinate origin.The errors between the numerical results and analytical solutions show a good stability,implying the reasonableness of the deduced formulas.
The programing sentence of Freefem++is characterized as concise,making the finite element software easy to be used for teaching and scientific research.Its disadvantage also exists for its no supplying gradient and normal programing elements in polar coordinate system,but merely Cartesian system.This drawback makes the software impossible to deal with physical problems in polar coordinate system.In order to tackle the problem,we derived the relation of gradient and normal direction between the two coordinate systems.We applied the transformation to Poisson equation in 1/4 circle domain,and scripted the program in the format of Freefem++.Results indicate that the numerical result of estimated function is consistent with the analytical solution,except those around the coordinate origin.The errors between the numerical results and analytical solutions show a good stability,implying the reasonableness of the deduced formulas.
引文
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[2]房毅,张先梅,钟菊花,等.基于“以学生为中心”的计算物理教学改革探索[J].物理通报,2017(10):8-10.
[3]彭芳麟,梁颖,忻蓓.计算软件在计算物理课程中的地位和作用[J].大学物理,2013(8):6-11.
[4]宋淑红,王双虎.基于微分形式的广义有限体积方法[J].应用数学学报,2016(3):382-391.
[5]杨波,徐义贤,何展翔,等.考虑海底地形的三维频率域可控源电磁响应有限体积法模拟[J].地球物理学报,2012(4):1390-1399.
[6]梁建文,朱俊.饱和软土场地中地下结构非线性地震响应分析的一个FEM-IBEM耦合方法[J].岩土工程学报,2018(11):1977-1987.
[7]何超,周顺华,狄宏规,等.饱和土-隧道动力响应的2.5维有限元-边界元耦合模型[J].力学学报,2017(1):126-136.
[8]尚月强.FreeFem++与偏微分方程有限元数值计算[J].贵州师范学院学报,2011(12):1-5.
[9]尚月强.基于MPI+FreeFem++的有限元并行计算[J].软件导刊,2012(2):27-29.
[10]杜世森.基于FreeFEM++的一阶有限元解法[D].长沙:湖南大学硕士论文,2011.
[11]姚端正,梁家宝.数学物理方法[M].第三版.北京:科学出版社,2017:1-341.
[12]梁昆淼.数学物理方法[M].第四版.北京:高等教育出版社,2010:1-448.