adaptive refinement in vibrational analysis and isogemetric analysis

7/30/2019 Adaptive Refinement in Vibrational Analysis and Isogemetric Analysis

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Adaptive Refinement inVibrational Analysis & New

Developments

ABHISHEK KUMAR

QHS 019


2/22

Principles of mesh refinement

Run the same model on grids with different space/time resolutions

Required for the embedding:

A time integration algorithm

Grids interactions

Required for the adaptivity:

A refinement criterion

An efficient grids initialization

procedure


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Error Indicators: This is the mechanism by which an adaptivemesh generator decides to coarsen or refine the mesh. The question we

need to know a prioriis what sorts of processes do we wish to track (error

indicators try to equi-distribute the error in a global sense). The errorindicator can do its job ifthe spatial and temporary resolutions have high

fidelity. This means that spatial, temporal, and boundary conditions should

have a balanced order of accuracy.

Parallelization/Domain Decomposition: Modifying the

data structures dynamically slows the computations. E.g., the domaindecomposition needs to be a direct by-product of the adaptive mesh

generator. A good first candidate for AM is statically adaptive grids where

the grid is modified and held fixed for the entire simulation. This must work

well before moving onto dynamically adaptive grids.

Some Standing Issues for AdaptiveMethods


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Coupling of Dynamics and Physics: Sub-grid scale parameterization isnotoriously inconsistent meaning that changing the grid resolutionchanges the results. Also, the dynamics must use the properapproximations for the smallest scales.. This has direct effects on the

time-integration strategies.

Handling Time-stepping: By definition, multi-scale problems haveprocesses occurring at all spatial scales that then require specialtreatment with respect to time in order to run efficient simulations (stiff andnon-stiff parts).

For explicit methods, the time-step has to be small enough to maintain stability.

For semi-implicit methods, one must still must adhere to the time-step stabilityof the nonlinear terms.

High-accuracy yet efficient time-integrators can be obtained. These methodsare easily modified to yield variable order in time with adaptive time-stepping.

Some Standing Issues for Adaptive Methods(Continued)


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R-refinement: The number of points remains constant for all time; however, theposition of these points move.

H-refinement: The number of grid points changes with time.

P-refinement: The number of grid points and their locations remain the same, thenumber of modes used to construct the solution changes. This is only possible withGalerkin methods.

HP-refinement: Combining AMR (h-refinement) and (p-refinement)

(e.g., Bernard, Chevaugeon, Legat, Deleersnijder, Remacle 2007).

K-refinement

Key Point: All these methods move points closer together so that special handlingof the time-step must be considered.

General Classification of Adaptive Methods


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6

Principles of mesh refinement

G0

G1

G2

interpolation

1

6

543

2

11

10

987 1312

update

Time integration algorithm


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Mode Dependent Error Estimator for

Vibration Analysis


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Spatial Error EstimatorMode Superposition Technique

Generalized coordinate transformation,

where is an n x m eigenvector (mode shape) matrix

The time history of dynamic stresses

where (t) = the stress components for Kth mode

The spatial error estimator at each time step in energy norm

where *(t) = improved/smoothed stress and

N = total number of elements

The error indicator at each time step

where is the exact energy norm at time t(sec)

Compute the element error indicators at the time step at which

is maximum

)t(qq)t(x km

1k

k

)t(qBDx(t)BD)t( km

1kk

21N

A

*1-* dA)t()t(D(t)-)t()t(e

100xu(t)

e(t)(%))( t

)t(u

)t(


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Explicit Time-Integrators

Semi-Implicit (IMEX) Time-Integrators

Lagrangian Time-Integrators

Fully-Implicit Time-Integrators

Time-Stepping Strategies


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Implementing Boundary Conditions

For explicit time-integrators,

implementing BCs is trivial (a

posteriori).

For semi-implicit/implicit time-integrators, the BCs have to be

incorporated into the time-integrator.

BCs can be generalized quite

naturally via Lagrange multipliers as

such: Where tau satisfies all BCs

including NFBC and NRBCs

We are currently extending high-

order BCs into this formulation.

q

t S(q) n

q

t S(q) q

n1 P

(BC)q


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Isogeometric Analysis

of StructuralVibrations

TJR Hughes

JA Cottrell


12/22

Independent evolution of CAD and FEM

CAD (NURBS) and Finite Elements evolved in different communities

before electronic data exchange

FEM developed to improve analysis in Engineering

CAD developed to improve the design process Information exchange was drawing based, consequently the

mathematical representation used posed no problems

Manual modelling of the element grid

Implementations used approaches that best exploited the limited

computational resources and memory available. FEA was developed before the NURBS theory

FEA evolution started in the 1940s and was given a rigorous

mathematical foundation around 1970 (e.g, ,1973: Strang and Fix's An

Analysis of The Finite Element Method)

B-splines: 1972: DeBoor-Cox Calculation, 1980: Oslo Algorithm
http://en.wikipedia.org/wiki/Gilbert_Stranghttp://en.wikipedia.org/wiki/George_Fixhttp://en.wikipedia.org/wiki/George_Fixhttp://en.wikipedia.org/wiki/Gilbert_Strang


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Why have NURBS not been used in FEA?

FEA was developed before the NURBS theory NURBS and Finite Elements evolved in different communities before

electronic data exchange

It was agreed that higher order representations in most cases

did not contribute to better solutions

Current computers have extreme performance compared to earliercomputers. Allows more generic solutions.

Mathematical representation in CAD and FEA chosen based on

what was computationally feasible.

We needed someone with high standing in FEA to promote the idea

of splines in analysis

Tom Hughes did this in 2005

The Computer Aided Design Community has adopted the idea

A new drive in spline research after 10 quite years


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The Isogeometric simulation process

Create geometry

CAD model

Translate model(can involve

simplification

and

approximaton)

Isogeometric

model

Add boundary

conditions,

properties

Refinement

(the model is

essencially the

same, only

enriched with

more information)

Isogeometric

mesh

Perform

analysis

Visualization

Result

Update geometry

(which version of the

geometry should be

updated?)

An analysis suitable model

USER


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Non-uniform rational B-spline

Non-uniform rational basis spline (NURBS) is amathematical model commonly used in computer

graphics for generating and representing curves

and surfaces which offers great flexibility and

precision for handling both analytic (surfacesdefined by common mathematical formulae) and

modeled shapes.

Development of NURBS made possible a

mathematically precise representation offreeform

surfaces like those used for ship hulls, aerospaceexterior surfaces, and car bodies, which could be

exactly reproduced whenever technically needed.

NURBS are commonly used in CAD, CAM, and

CAE. NURBS tools are also found in various 3D

modeling and animation software packages.


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Why are splines important to Isogeometricanalysis?

Splines are polynomial, same as Finite Elements

B-Splines are very stable numerically

B-splines represent regular piecewise polynomial structure in a

more compact way than Finite Elements

NonUniform rational B-splines can represent elementary curves andsurfaces exactly. (Circle, ellipse, cylinder, cone)

Efficient and stable methods exist for refining the piecewise

polynomials represented by splines

Knot insertion (Oslo Algorithm, 1980, Cohen, Lyche, Riesenfeld)

B-spline has a rich set of refinement methods


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Adaptive MeshApplications


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Adaptive Applications

Highly adaptive and irregular applications

Amount of work per task can vary

drastically throughout the execution Computations in the interesting regions

of the domain larger than for otherregions

It is difficult to predict which regions willbecome interesting


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AMR Applications

An example of such applications is ParallelAdaptive Mesh Refinement (AMR) for multi-scale applications

Adaptive Mesh Mesh or grid size is notfixed as in Laplace/Jacobi, but interestingregions are refined to form finer level

grids/mesh E.g.: to study crack growth through a

macroscopic structure under stress


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AMR Applications Crackpropagation

Such a system is subject to the laws ofplasticity and elasticity and can besolved using finite element method

Crack growth forces the geometry ofthe domain to change

This in turn necessitates localizedremeshing.


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AMR Applications- Adaptivity

Adaptivity arises when advances crossesfrom one subdomain to another

It is unknown in advance when or wherethe crack growth will take place and whichsubdomains will be affected

The computational complexity of a

subdomain can increase dramatically due togreater levels of mesh refinement

Difficult to predict future workloads


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adaptive refinement in vibrational analysis and isogemetric analysis

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