This document describes an efficient linear elastic finite element method (FEM) solver that uses automatic local grid refinement and accuracy control. The solver achieves the following objectives: 1. Imposes a controlled accuracy level prior to solving. 2. Provides a fast solver with a quasi-minimal problem size. 3. Includes an automatic 3D mesher. It uses an algorithm based on multiple discretization levels, leading to a locally adaptive multigrid solver. Local mesh refinement is based on error estimators and uses non-conforming meshes. The method aims to provide an efficient solver with linear computational complexity.

1. AN EFFICIENT LINEAR ELASTIC
FEM SOLVER USING AUTOMATIC
LOCAL GRID REFINEMENT AND
ACCURACY CONTROL
HARSHAL PATIL
1 3ME63R26

2. INTRODUCTION
Industrial problems with higher degrees of freedom.
Features
A controlled accuracy level which is imposed priori.
A fast solver & a quasi -minimal problem size.
A n a u t o m a t i c 3 D m e s h e r.
To a c h i e v e t h e s e o b j e c t i v e s
An algorithm based on several discretization levels is
u s e d , l e a d i n g t o a l o c a l l y a d a p t i v e m u l t i g r i d s o l v e r.

3. 
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Discretization Error
Optimal Mesh Design
Error Estimators
The dynamic mesh construction based on local or
global error estimators is the basic concept of
adaptive techniques.
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h- method
P- method
hp- method
In this work,
h-method with local error indicators is used

4. Dual nature of multigrid techniques
 Ef ficient Solvers
 Error Estimators
Combination of 2 multigrid technique advantages
 An ef ficient convergence of numerical errors
 Simple & intrinsic error estimation.
Non-conforming techniques are found to be more
efficient.
Local mesh refinement also uses non -conforming
meshes.

5. MULTIGRID TECHNIQUES
Intensive use of several discretization levels
Ef f icient communication between these dif ferent
levels
The per formance gain is conditioned by a limited
computing cost for this interlevel communication.
The algorithm presented in this paper merges an
accuracy control, an ef ficient multilevel solver & an
a u t o m a t i c 3 D m e s h e r.

6. Several error estimators are available
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Equilibrium Unbalance
Error in the Constitutive Relation
Stress field non-regularity
For static linear calculations, displacement & element
energy criteria are well suited.
All the results presented below are based on an
element energy criterion.

7. EFFICIENCY
The method efficiency is based on
 An ef ficient solver
 A localized problem
 A dedicated implementation

8. MULTIGRID SOLVER
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The main goal is to obtain a solver with complexity
close to 1 (optimal case)
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Complexity is defined as the relation between the
problem size and the number of numerical
operations required for the solution.
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Low frequencies on a given fine mesh appear as
high frequencies on a much coarser mesh and can
be ef ficiently corrected

9. MULTIGRID SOLVER

10. LOCAL REFINEMENT
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Coarse problem is refined in cer tain par ts of the
domain based on accuracy criterion.
The circled fine node is corrected using the nearest
node of the FINE SKIN MESH.
I d e a l l y, o n e s h o u l d b e a b l e t o r e f i n e t h e s k i n m e s h
i f n e c e s s a r y.

11. IMPLEMENTATION
Keep the execution time complexity as linear as
possible using minimum memory
The preferred data structure is a compressed format
An internal boundar y on level (L) is created between
the refined and non refined domain

12. RESULT

13. CONCLUSION