ﯾﮏ اﯾﺪه ﺟﺪﯾﺪ ﺑﺮ اي اﻋﻤﺎل ﺷﺮاﯾﻂ ﻣﺮزي اﺳﺎﺳﯽ در روش ﺑﺪون اﻟﻤﺎن ﮔﺎﻟﺮﮐﯿﻦ ﺑﺮايﺣﻞ ﻣﻌﺎدﻻت ﺑﺎ ﻣﺸﺘﻘﺎتﺟﺰﺋﯽ ﺑﯿﻀﻮي

A new approach to apply the essential boundary conditions in element free Galerkin method for elliptic partial differential equations

روش ﺑﺪون اﻟﻤﺎن ﮔﺎﻟﺮﮐﯿﻦ ﯾﮏ روش ﺷﻨﺎﺧﺘﻪ ﺷﺪه ﺑﺮ اي ﺣﻞ ﻣﻌﺎدﻻت ﺑﺎ ﻣﺸﺘﻘﺎت ﺟﺰﺋﯽ اﺳﺖ. اﻋﻤﺎل ﺷﺮاﯾﻂ ﻣﺮز ي اﺳﺎﺳﯽ در اﯾﻦ روش ﮐﻪ ﺑﺮ اﺳﺎس ﺗﻘﺮﯾﺐ ﮐﻤﺘﺮﯾﻦ ﻣﺮﺑﻌﺎت ﻣﺘﺤﺮك اﻧﺠﺎم ﻣﯽﺷﻮد، ﺑﺎ ﭘﯿﭽ ﯿﺪﮔﯽﻫﺎﯾﯽ ﻫﻤﺮاه اﺳﺖ. از آﻧﺠﺎ ﮐﻪ ﺗﻮاﺑﻊ ﺷﮑﻞ ﺗﻘﺮﯾﺐ ﮐﻤﺘﺮﯾﻦ ﻣﺮﺑﻌﺎت ﻣﺘﺤﺮك در ﺧﺎﺻﯿﺖ دﻟﺘﺎي ﮐﺮوﻧﯿﮑﺮ ﺻﺪق ﻧﻤﯽﮐﻨﻨﺪ، ﻧﻤﯽﺗﻮان ﻫﻤﺎﻧﻨﺪ روش ﻋﻨﺎﺻﺮ ﻣﺘﻨﺎﻫﯽ، ﺷﺮاﯾ ﻂ ﻣﺮزي اﺳﺎﺳ ﯽ را ﺑﻪ ﺻﻮرت ﻣﺴﺘﻘﯿ ﻢ در ﻓﺮم ﺿﻌﯿﻒ ﮔﺎﻟﺮﮐﯿﻦ ﻣﻌﺎدﻟﻪ اﻋﻤﺎل ﮐﺮد و ﻧﯿﺎز ﺑﻪ روشﻫﺎي اﺻﻼﺣﯽ ﺑﺮاي ﻓﺮم ﺿﻌﯿﻒ ﻣﻌﺎدﻟﻪ دارﯾﻢ. در اﯾﻦ ﻣﻘﺎﻟﻪ ﯾﮏ اﯾ ﺪه ﺟﺪﯾﺪ ﺑﺮاي اﻋﻤﺎل ﺷﺮاﯾﻂ ﻣﺮزي اﺳﺎﺳﯽ در روش ﺑﺪون اﻟﻤﺎن ﮔﺎﻟﺮﮐﯿﻦ ﺑﺮ اي ﺣﻞ ﻣﻌﺎدﻻت ﺑﺎ ﻣﺸﺘﻘﺎت ﺟﺰﺋﯽ ﺑﯿ ﻀﻮي ﻣﻌﺮﻓﯽ ﻣﯽﺷﻮد. اﯾﻦ اﯾﺪه ﺑﺮاﺳﺎس روش ﮐﻤﺘﺮﯾﻦ ﻣﺮﺑﻌﺎت ﻣﺘﺤﺮك دروﻧﯿﺎب اﺳﺖ. در اﯾﻦ روش اﺑﺘﺪا ﺷﺮاﯾﻂ ﻣﺮزي را در ﺗﻘﺮﯾﺐ ﮐﻤﺘﺮﯾﻦ ﻣﺮﺑﻌﺎت ﻣﺘﺤﺮك ﺗﺎﺑﻊ اﻋﻤﺎل ﻣﯽﮐﻨﯿ ﻢ ﺳﭙﺲ ﺗﻘﺮﯾﺐ ﺣﺎﺻﻞ را در روش ﺑﺪون اﻟﻤﺎن ﮔﺎﻟﺮﮐﯿﻦ ﺑﻪ ﮐﺎر ﻣﯽﺑﺮﯾﻢ. ﺑﻨﺎﺑﺮاﯾﻦ ﺷﺮاﯾ ﻂ ﻣﺮزي ﺑﻪ ﺻﻮرت ﻣﺴﺘﻘﯿﻢ اﻋﻤﺎل ﻣﯽﺷﻮد. در اﯾﻦ ﻣﻘﺎﻟﻪ اﺑﺘﺪا ﺗﻘﺮﯾﺐ ﮐﻤﺘﺮﯾﻦ ﻣﺮﺑﻌﺎت ﻣﺘﺤﺮك دروﻧﯿﺎب ﻣﻌﺮﻓﯽ ﻣﯽﺷﻮد و ﺳﭙﺲ ﻧﺤﻮه اﻋﻤﺎل ﺷﺮاﯾ ﻂ ﻣﺮزي ﺑﯿﺎن ﺧﻮاﻫﺪ ﺷﺪ. در اﻧﺘﻬﺎ ﺑﺎ اراﺋﻪ ﭼﻨﺪ ﻣﺜﺎل ﻣﺨﺘﻠﻒ ﮐﺎراﯾﯽ روش را ﻧﺸﺎن ﻣﯽدﻫﯿﻢ .

The element free Galerkin method is a well-known method for solving partial differential equations. Applying essential boundary conditions in this method, that based on moving least squares approximation, have some complexities. Since the shape functions of the moving least squares approximation do not satisfy the property of Kronecker delta function, therefore imposing essential boundary conditions is not as trivial as in the finite element method and we need some modifications of the Galerkin weak form of the equation. In this paper we propose a new approach to apply essential boundary conditions in element free Galerkin method for solving elliptic PDEs. This approach is based on interpolating moving least square method. First we apply the essential boundary conditions in the moving least square approximation of the function then the approximation is used in element free Galerkin method. Thus the essential boundary condition is applied directly. In this paper we first introduce the interpolating moving least squares approximation, and then describe how to apply the boundary conditions. Finally, some different examples show the accuracy and efficiency of the method.