Natural Element Method in Solid Mechanics Rajagopalachary

8/10/2019 Natural Element Method in Solid Mechanics Rajagopalachary

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Natural Element Method

in Solid Mechanics

T. Rajagopalachary andV.V. Kutumbarao

Metallo 2007(Commemorative Conference on 80th

Birthday of Professor T. R. Anantharaman)

IIT, KanpurDecember 9, 2007


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Natural Element Method in Solid Mechanics

Finite element Method Mesh-free methods

1. Direct solution of a set of Pde and

associate boundary/initial conditions orfrom its variational or weak form isattempted

Galerkin formulations are quite popular.

Several variants from fully pointcollocation methods to very similar toFEM or FDM

2. Discretize domain into elements andnodes. Prescribed connectivity between anelement and its associated nodes.

2. The continuum is discretized to haveinternal and boundary nodes. The solutionat nodal points is approximated

3. Lagrangian or Labatto polynomials, etc)

provide element wise local approximationof solution

3. Approximants ( not necessarily

interpolants ) are used e.g. Radial basisfunctions, multiquadrics, from MLSapproximation

4. Interpolation and test functions usedhave properties of Partition of Unity andKronecker delta, are often consistent

They may lack the properties of Partition ofunity and /or Kronecker delta

5. Element equations are assembled usingnodal connectivity. Enforcing boundarycondition is easy

6. Assembly is node-wise. Imposition ofessential boundary conditions is not trivial.Several procedures are suggested

6. Solution of algebraic equations give thefiled variables of the problem at nodes

6. Solution of algebraic equations give thefiled variables of the problem at nodes.


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Meshfree methods differ from FEM in

Construction of interpolants or approximants

Numerical quadrature and matrix assembly

Imposition of boundary conditions Post-processing

NEM uses natural neighbour coordinates or natural neighbour interpolants

Sibson

non-Sibson

Characteristics of NEM

NEM shape functions are interpolants and are positive definite

NEM interpolants obey partition of unity and reproduce linearvariations (A requirement for convergence)

NEM shape functions are infinitely smooth except at the point ofdefinition Can be used in solving higher order pde than quadratic.

Where C1 Continuity is a must natural neighbor coordinates are

incorporated into Bezier splines.


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Basic steps in NEM

Discretize the domain for nodes

Construct the weak form of PDE

Assume NEM interpolants and Formulate

NEM discrete matrix equations

Construct a background cellular network.

Loop over a cell

Determine the gauss point coordinates

To impose Dirichlet conditions for non-

convex domains with Sibson interpolation

use any of

Lagrange multipliers method

Penalty function method.

Transform method

Solve algebraic equations. Coefficient

matrices of NEM are sparse and symmetric

though not banded.

Post processing in NEM is simple.


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Theoretical formulation of typical problem in solid mechanics

Constrained Principle of virtual work (weak form)

where B is the strain displacement tensor,

Dis the elastic stiffness tensor

,b is the body force vector,

displacement constraint is

Lagrange multipliers vector is is initial strain tensor

duCduCdDBdbdBDB TT

T

IIIJ

T

I )()(*

0)(uC

*,

T


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NEM trial functions are defined by

Test and trial functions are the same.

Key ingredients of all meshfree methods, which differ from finiteelement based procedure in detail, are

( i ) Calculation of nodal shape functions,

(ii) Evaluation of matrices formulations by a Galerkin procedure,

(iii) Imposition of boundary conditions.

(iv) Post processing of results

Iu

n

I Ixhu

1)(

Theoretical formulation (continued)


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Construction of NEM shape functions is based on Voronoi diagrams

The Voronoi construction is formed by

drawing perpendicular bisectorsbetween an arbitrary given node and

other nodes . When we consider all

the other nodes ( ), the

perpendicular bisectors of lines

constitute a closed polygon around

node . This closed convex polygon is

called as Voronoi cell.

In

In

Jn

JI nn

JInn

In


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The triangulation becomes Delaunay if for any arbitrary triangle,

the circumcircle of the triangle contains no other vertex than itsthree vertices. Delaunay triangulation is also called as Delaunay

tessellation or Delaunay graph.

Line dual to the Delaunay triangulation is the Voronoi diagrams.

Delaunay triangle is the line dual of Voronoi diagram


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.

Natural neighbors of a node are thosewhose Voronoi cell shares a common edge.

Node A has as natural neighbours in B,C

and D but not the nodes E or F. Euclidean

distances of nodes A and E or A and F are

smaller than the those between nodes A and

D or A and C


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(a) Original Voronoi diagram and x

(b) First and second order Voronoi cells about x

Construction of First and second order Voronoi diagrams of point


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Natural Neighbor based Interpolants

Sibson Interpolant Laplace or Non-Sibsonian

Interpolant


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NEM Shape functions on a regular domain of support

(a) Domain of support, (b) Sibson Interpolant, (c) Laplace interpolant


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Issues connected with NEM applications in Solid Mechanics

Numerical quadrature of integrals

Imposition of boundary conditions


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A patch of elements on unit square containing 70 nodes

2L 1H

Different nodal discretizations for a unit square for Patch test

Patch of elements with regular

spacing containing 25 nodes

Patch test with 8 nodes

per unit square


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Error norm calculations for Patch test

2L 1HError Norm Error norm

Number of nodes in

unit square plate

NEM FEM NEM FEM

8 7.6125e-03 3.72e-17 6.8265e-02 2.98e-16

25 6.1010e-04 8.76e-17 6.4932e-03 5.68e-16

70 3.6413e-03 1.87e-16 1.55e-01 2.04e-15


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NEM Compared

Finite elementmethod

Natural elementmethod

Element free Galerkin

Interpolants are used Interpolants are used Approximants are used. It isconstructed solely from a setof notes and weightfunctions

Interpolants havecompact support ofshape functions

Interpolants havecompact support ofshape functions

Approximants have compactsupport of shape functions

Integrals in weak formare estimated to verygood precision

Shape functions are notpolynomials; Numericalintegration is an issue

The precision of Numericalintegration is a much greaterissue.

Coefficient matrix is

banded

Not banded Not banded

Computational costs arelow

costs are higher Costs are higher

Mesh distortion andentanglement is an issue

Not an issue. Not an issue.


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Thank You

Documents

Natural Element Method in Solid Mechanics Rajagopalachary