meshfree

CAD to Analysis Transition, IGA, GIFT, fracture and shape optimisation and error estimation short course notes _ CISM Short Course

Link to the slides and hand-written notes: http://hdl.handle.net/10993/31585

Stéphane Bordas

Plenary talk at XDMS2017
http://orbilu.uni.lu/handle/10993/31487
2017 Short course Lecture Notes
http://hdl.handle.net/10993/31585

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General query regarding MLPG method

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I have few general queries regarding MLPG meshfree method

1. Can the trial and test functions be taken same for solving a problem

2. In the problem if there are zones with different properties, for example different Coeffiencient of Permeability in the aquifers (Plan view-wise), how to handle the interface between the zones and assemble the global matrix

Meshfree Analysis of Elastic Bar

This may be usefull for learning meshfree methods

Meshfree Analysis of Elastic Bar

regards.

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Hi, everyone.

I'm going to employ the meshfree method to analyze composite model.

But, I can't understand how to determine the support radius although I have read some references.

I will use the circular support domain.

Need help with understanding Thin Plate Splines as Radial Basis function for Meshfree simulation

I want to try applying TPS as the RBF for the simulation of groundwater flow and transport. but the i'm getting error because of zeros in diagonal elements in the global matrix. Please help with the shape function

3. Meshfree methods and the patch test

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We are in the framework of small-strain two-dimensional linear elasticity without any body forces. Consider a domain that is discretized by a union of triangles and/or quadrilaterals (`patch of elements').  For C0 conforming approximations such as triangular/quadrilateral finite elements, the finite element approximation can exactly reproduce an arbitrary linear displacement field. Hence, if the exact solution is linear, then the finite element solution must match (within machine precision) the exact solution. In simple terms, passing the patch test for linear elasticity with standard conforming finite elements provides verification of one's implementation and is used to assess the same when new elements are proposed. For conforming elements, it is a sufficient condition for convergence (2nd order PDEs), and hence is the first problem that is solved when a new element/method is proposed. To carry out the patch test, the following steps are performed:

2. Is a mesh required in meshfree methods?

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In meshfree (this is more in vogue than the term meshless) methods, two key steps need to be mentioned: (A) construction of the trial and test approximations; and (B) numerical evaluation of the weak form (Galerkin or Rayleigh-Ritz procedure) integrals, which lead to a linear system of equations (Kd = f). In meshfree Galerkin methods, the main departure from FEM is in (A): meshfree approximation schemes (linear combination of basis functions) are constructed independent of an underlying mesh (union of elements).

However, since a Galerkin method is typically used in solid mechanics applications, (B) arises and the weak form integrals need to be evaluated. Three main directions have been pursued to evaluate these integrals:

Where can I read about the basic ideas of the meshfree methods?

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For someone with a background in solid mechanics and finite element methods, where should he go to read up on the elementary ideas of the meshfree methods?