9351

8633

كارشناسي ارشد

علوم رياضي

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

1393

نه،71ص.: مصور،جدول،نمودار

ص.ع.به فارسي و انگليسي

مهدي تاتاري

16/9/93

داود ميرزائي، محمدرضا رئوفي

رياضي

ID8633

Computational Techniques in Implementation of some Meshless methods Mohammad Damavandi Nia mt damavandi@math iut ac ir Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Mahdi Tatari mtatari@cc iut ac ir Advisor Dr Amir Hashemi amir hashemi@cc iut ac ir 2010 MSC 65M99 65D17 Keywords Reproducing kernel particle method Galerkin method Meshless methods Kd tree algorithm Numerical integration MATLAB programming AbstractIn this thesis the Galerkin method is used for solving PDEs with the aid of RKPM meshless methodfor approximations Penalty method is used for enforcing Dirichlet boundary conditions The timeand memory usage are tried to be improved in di erent problems using programming techniques e g vectorization using kd tree for nearest neighbour search using Gauss elimination method for matrixinversion and choosing the correct kinds of functions to de ne to name a few The results obtainedfrom solving 2D and 3D elliptic problems e g for Laplace and elasticity problems indicate the highaccuracy and speed of the peresented techniques in solving equations It is also possible to preventmesh locking in the use of the penalty method A Meshless approach based on a Reproducing Kernel Particle Method is developed for metal forminganalysis The displacement shape functions are constructed using the reproducing kernel approxima tion that satis es consistency conditions The variational equation of materials with loading pathdependent behavior and contact conditions is formulated with reference to the current con gura tion Numerical methods are crucial for an accurate simulation of physical problems as the partialdi erential equations describing them usually require approximation schemes for their solution Ap proximation is necessaryeither for the complexity of the equations and or for the complexity of thegeometry of de nition of these equations Among all the available numerical schemes meshbasedmethods have become the most popular tools for engineering analysis over the last decades in aca demic and industrial applications The most conventional meshbased numerical method is the FiniteElement Method FEM well known as the most thoroughly developed method in engineering FEM isnowadayswidely used by engineers in all elds and several well assessed commercial codes are available However in FEM it is very complicated to model the breakage into a large number of fragments asFEMis intrinsically based on continuummechanics where elements cannot be broken The elementsthus have to be either totally eroded or stay as a whole piece but this leads to a misrepresentationof the fracture path serious errors can also occur because the nature of the problem is nonlinear Toovercome these problems related with the existence of a mesh a numerical scheme that relies onlyon nodes would be highly bene cial These methods are called meshfree or meshless since they donot need a mesh to construct the approximationof the solutionof a givendi erential equation Novelnumerical methods known as Meshless Methods or Meshfree Methods and in a wider perspective Partition of Unity Methods promise to overcome most of disadvantages of the traditional nite ele ment techniques The absence of a meshmakes meshfree methods very attractive for those problemsinvolving large deformations moving boundaries and crack propagation However meshfree meth ods still have signi cant limitations that prevent their acceptance among researchers and engineers namely the computational costs This paper presents an indepth analysis of computational techniquesto speedup the computation of the shape functions in the Reproducing Kernel Particle Method and

مهدي تاتاري

داود ميرزائي، محمدرضا رئوفي