07 modelisation_ondes

SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 1

Semaine 7 Modélisation de la propagation d’ondes

guidées

Ramy Mohamed

Été 2011


A little bit of Philosophy

Physical World

Mathematical World

Engineering (Numerical World)


Plan

• Objectives of session: • What is special about guided waves, both physically and numerically.

• Identify the major Numerical Methods that is used in modeling guided waves propagation.

• Understand the major factors controlling the accuracy and validity of a numerical simulation of a guided waves propagation

• Organisation of session: • Strong Form schemes (Finite Difference & Spectral Schemes)

• Break

• Weak Form schemes ( Finite Element & Spectral Element)

• Devoir


Reminder

• At the beginning there was ODE

Given , Find a differentiable function defined for (a) (b) For all In order to have a solution, should be continuous in time and bounded in u, L > 0


Discretization

Exact Approximate

We need to achieve a prescribed accuracy with the minimum number of function evaluations.


Discretization

• Types of Errors: Discretization errors: Numerical approximation

issue. Round-off errors: Implementation issue

• Numerical Scheme Attributes • Accuracy, Stability and Convergence


FD Schemes

• Forward Operator

• Backward Operator

• Central


– Local approximation error – Exact solution – Approximate solution

Order of accuracy The numerical scheme has order of accuracy if

Accuracy


Stability of FD Schemes

• Lax-Richtmyer Stability (zero Stability) as in bounded interval [0, t] • Absolute Stability


Stability Sensitivity of the algorithm to small perturbations in input (error driven).

–Numerical Stability –Algorithmic Stability


Convergence of Explicit FD Schemes

FD Scheme is Consistent if Convergence = Stable time step + Consistency


FD Schemes

Leapfrog (2nd Order)

Crank-Nicolson (2nd Order)

Runge-Kutta

(2th Order)


Polynomial Approximation

The unknown solution Is Approximated by

Lagrange Polynomials

– An example of Algorithmic stability

– Barycentric-Remerez Algorithm is more stable


Differentiation Matrices

Interpolation Spectral

(Global Support)

FD

(local Support)

FD FD

Spectral


Spectral Schemes

• To overcome Runge Phenomenon

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1x 10

-3 equispaced points

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

equispaced points

-1 -0.5 0 0.5 1-4

-2

0

2

4x 10

-5 Chebyshev points

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

Chebyshev points


Spectral Schemes

100 101 102 103 10410-15

10-10

10-5

100

N

erro

r

Convergence of fourth-order finite differences

N-4

100 101 10210-15

10-10

10-5

100

N

erro

r

Convergence of spectral differentiation


Prelude to Guided Waves Simulation

• Types of PDEs: Elliptic : Time Independent, Equilibrium

phenomena. Parabolic : Time dependent, and diffusive Hyperbolic : Time dependent, wavelike, with a

finite speed of propagation


Prelude to Guided waves Simulation

Hyperbolic Parabolic


Wave Equation in 1D

Strong Form: find such that • Governing Equation

• Initial Conditions

• Boundary Conditions Essential Natural


FD Schemes for 1D Wave Equation

2nd Order Temporal-Spatial Scheme: Courant Friedrichs Lewy (CFL) Condition:


FEM Boundary Conditions

• Essential Boundary Condition At least one point of the boundary should have an essential BC.

• Natural Boundary Condition

Common nodes satisfy the essential BC.


FEM Static

• System of equations: strong form, difficult to solve, over complex geometry.

• Weak form: requires weaker continuity on the dependent variables.

• Weak form is often preferred for obtaining an approximated solution.

• Formulation based on a weak form leads to a set of algebraic system equations – FEM.

• FEM can be applied for practical problems with complex geometry and boundary conditions.


FEM Static

•Exact at the nodes •There exists at least one point in each element at which the derivative is exact •The derivative is 2nd order accurate at the midpoints of elements

x

F ( x )

nodes elements

Unknown function of field variable

Unknown discrete values of field variable at nodes


Preprocessing

FEM Static

• Step 1: Domain discretization (Meshing) • Step 2: Displacement interpolation (Element Order) • Step 3: Formation of FE equation in local coordinate system • Step 4: Coordinate transformation or mapping • Step 5: Assembly of FE equations • Step 6: Imposition of displacement constraints • Step 7: Solving the FE equations

Solver


FEM Static Step 1

• The solid body is divided into N elements with proper connectivity – compatibility.

• All the elements form the entire domain of the problem without any overlapping – compatibility.

• There can be different types of element with different number of nodes. (Conforming vs Non-Conforming Elements)

• The density of the mesh depends upon the accuracy requirement of the analysis.

• The mesh is usually not uniform, and a finer mesh is often used in the area where the displacement gradient is larger.


• 2D solid elements are applicable for the analysis of plane strain and plane stress problems.

• A 2D solid element can have a triangular, rectangular or quadrilateral shape with straight or curved edges.

• A 2D solid element can deform only in the plane of the 2D solid.

• At any point, there are two degrees of freedom (dofs) in the x and y directions for the displacement as well as forces.

FEM Static Step 2


FEM Static Linear Triangular Element

•Less accurate than quadrilateral elements •Used by most mesh generators for complex geometry (Unstructured Meshing) •A linear triangular element:

x, u

y, v

1 (x1, y1) (u1, v1)

2 (x2, y2) (u2, v2)

3 (x3, y3) (u3, v3)

A fsx

fsy


FEM Static Linear Rectangular Element

•Non-constant strain matrix •More accurate representation of stress and strain •Regular shape makes formulation easy

x, u

y, v

1 (x1, y1) (u1, v1)

2 (x2, y2) (u2, v2)

3 (x3, y3) (u3, v3)

2a

fsy fsx

4 (x4, y4) (u4, v4)

2b

η

ξ


FEM Static Element Distortion

• Use of distorted elements in irregular and complex geometry is common but there are some limits to the distortion.

• The distortions are measured against the basic shape of the element – Square ⇒ Quadrilateral elements – Isosceles triangle ⇒ Triangle elements – Cube ⇒ Hexahedron elements – Isosceles tetrahedron ⇒ Tetrahedron elements



• Aspect Ratio Distortion

b

a

3 Stress analysis10 Displacement analysis

ba

≤

Rule of thumb:



•Angular Distortion

skew Taper b a

b<5a

<120°

>60°


FEM Static Increasing Accuracy: h adaptation


FEM Static Increasing Accuracy: p adaptation


FEM for Guided Waves Transient Analysis

We need To march the system in time (find the solution at discrete time steps a) Accuracy b) Stability


Time Integration Transient Analysis

• The direct integration method is basically using the finite difference scheme for time marching.

• There are mainly two types of direct integration method: implicit and explicit.

• Implicit method (e.g. Newmark’s method) is more efficient for relatively slow phenomena.

• Explicit method (e.g. central differencing method) is more efficient for fast phenomena, such as impact, explosion and guided waves.


Transient Analysis

Central Difference Method

1) Explicit 2) A special initialization procedure is needed


Transient Analysis

1- Identify the frequencies contained in the loading.

2- Choose a FE mesh that can accurately represent the static, and accurately all frequencies up to about

3- Perform a Direct time integration

analysis.


Transient Analysis Amplification factor Vibration Period T


Transient Analysis


Transient Analysis

Numerical Dispersion: Higher frequency modes propagate numerically, while not representing a physical phenomena


Time Integration Transient Analysis

• Time Step (CFL Condition)

• For 1% Numerical Dispersion 20 Points per minimum Wavelength


Spectral Element

Mapping (Superparametric)


Spectral Element


Numerical Dispersion


CPU time


Spectral Element

S0 interaction with Notch


Spectral Element

A0 interaction with notch


SEM vs FEM


Spectral Element Time stability FEM mesh (quartic Quad)


Spectral Element Time stability SEM mesh (Quad 3X6)


Thanks

Questions


Types of PDEs

2

0( 0

0 - 4ac)

Ellipticb Parabolic

Hyperbolic

< = >

07 modelisation_ondes

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