CEE 235B Final Project_John_Wick

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CEE 235B ADVANCED FINITE ELEMENT ANALYSIS OF STRUCTURES

Author: John Wick

UCLA MASTERS in ENGINEERING

Field of Study: Mechanics of Structures

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TABLE OF CONTENTS

ACRONYMS ............................................................................................................................... v

1 ABSTRACT..................................................................................................................... 1

2 INTRODUCTION ............................................................................................................. 1

3 METHODS ...................................................................................................................... 2

4 RESULTS ..................................................................................................................... 11

5 DISCUSSION ................................................................................................................ 16

6 SUMMARY & CONCLUSIONS ..................................................................................... 22

7 REFERENCES .............................................................................................................. 23

APPENDIX A. MATLAB CODE .............................................................................................. A-1

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LIST OF FIGURES

Figure 1. Elastic Bar Impacting rigid wall @ Velocity V_0 ........................................................... 1

Figure 2. Matched method rationale ............................................................................................ 8

Figure 3. Approximate solution of Displacement vs Time .......................................................... 11

Figure 4. Approximate Solution of Stress vs Time ..................................................................... 12

Figure 5. Exact Solution of Displacement vs Time .................................................................... 13

Figure 6. Exact Solution of Stress vs time ................................................................................. 13

Figure 7. Comparison of displacement approximations to exact solution................................... 14

Figure 8. Comparison of stress approximations to exact solution .............................................. 15

Figure 9. Effect of element size on approximation ..................................................................... 15

Figure 10. Effect of time step size on approximation ................................................................. 16

Figure 11. Unstable finite element model results ....................................................................... 19

Figure 12. Damping of high frequency oscillations in approximate solution ............................... 20

Figure 13. L2 Error norm of Displacement ................................................................................. 21

Figure 14. H1 Error Norm of Stress ........................................................................................... 21

LIST OF EQUATIONS

Equation 1. 3D Wave equation .................................................................................................... 2

Equation 2. Strong form of the boundary value problem .............................................................. 2

Equation 3. Constitutive Law ...................................................................................................... 2

Equation 4. Strain Displacement Relationship ............................................................................. 2

Equation 5. Stress Strain Relationship in 1D ............................................................................... 3

Equation 6. Strain Displacement Relationship in 1D ................................................................... 3

Equation 7. Strong form of 1D elastic bar BVP ............................................................................ 3

Equation 8. Weak form of 1D elastic bar BVP ............................................................................. 4

Equation 9. Galerkin formulation of 1D elastic bar BVP ............................................................... 4

Equation 10. Finite element approximation of trial and weight functions in space and time ......... 5

Equation 11. Mass and Stiffness matrices ................................................................................... 5

Equation 12. Element Mass and Stiffness matrices ..................................................................... 5

Equation 13. Semi-Discrete equation of 1D elastic bar BVP........................................................ 6

Equation 14. Full discrete equation of 1D elastic bar BVP ........................................................... 6

Equation 15. Newmark Time integration method ......................................................................... 7

Equation 16. Ratio of temporal and spatial integration frequency to exact frequency .................. 8

Equation 17. Stability analysis of Newmark Time Integration ...................................................... 9

Equation 18. Critical Time step for central difference method ...................................................... 9

Equation 19. L2 error norm ....................................................................................................... 10

Equation 20. H1 error norm ....................................................................................................... 10

Equation 21. Rate of Convergence ........................................................................................... 10

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ACRONYMS

Acronym Description

1D 1 dimensional

2D 2 dimensional

3D 3 dimensional

BVP Boundary Value Problem

Det Determinant

DOF Degree of Freedom

E*I Stiffness (E = Elastic Modulus; I=rotational inertia)

Eq Equation

Hz Hertz

K Stiffness

L Length

NEN Number of Element Nodes

ODE Ordinary Differential Equation

PDE Partial Differential Equation

RMS Root Mean Square

Wn Natural Frequency

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1 ABSTRACT

The objective of this paper is to study the numerical characteristics of explicit and implicit time

integration methods and their stability and accuracy.

2 INTRODUCTION

This paper will guide the reader through the procedure of solving for displacement and stress

histories at various locations along a finite length elastic bar that impacts into a rigid wall with an

initial velocity V_0 as shown in figure 1 below.

Figure 1. Elastic Bar Impacting rigid wall @ Velocity V_0

The bar is made of a linear elastic material with constant material properties throughout its

length. The bar will be divided up into a linear finite element model consisting of 2-node

elements. Two methods will be described and developed to solve for the displacement and

stress histories during the impact event:

1. The implicit time integration method will utilize the central difference method with a

lumped mass

2. The explicit time integration method will utilize the average acceleration method with

consistent mass.

The methods section below will provide details in how the two different methods are developed

and utilized to calculate the history plots.

Due to the nature of a finite element analysis, it’s an approximation of a true solution. The error

of the approximation is an important factor in understanding how accurate a model is to the true

solution. The accuracy and stability of both methods will be discussed as they relate to the true

solution. Additional discussion will include how element size and time step size affect the

accuracy and stability of the numerical approximation.

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3 METHODS

This section will describe the process by which the governing 3 dimensional differential equation

is manipulated until a solution of displacement and stress at various locations along the length

of the bar is reached. The calculations of error between the approximate and exact solution will

also be described in this section.

Equation 1 below is the 3 dimensional wave equation that governs the bar’s response.

Equation 1. 3D Wave equation

Below are the boundary conditions and initial conditions that, coupled with equation 1, make up

the strong form of this boundary value problem.

Equation 2. Strong form of the boundary value problem

In order to reduce this to a 1 dimensional problem, the following relationships are substituted

into the strong form presented above.

Equation 3. Constitutive Law

Equation 4. Strain Displacement Relationship

When i=j=k=1, the constitutive law and strain displacement operations are reduced to the

following 1D equations

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Equation 5. Stress Strain Relationship in 1D

Equation 6. Strain Displacement Relationship in 1D

Utilizing equations 5 and 6, the strong form of the 1D elastic bar problem is show below

Equation 7. Strong form of 1D elastic bar BVP

In order to arrive at the semi and fully discrete equation from where the accelerations and

displacement are calculated and iterated, modify the strong form must be modified into the weak

form by multiplying it by an arbitrary weight function, wi and integrating over the entire space

and time domain.

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Equation 8. Weak form of 1D elastic bar BVP

After integrating by parts the second derivative of u with respect to x, the weak form integral

simplifies into 2 terms shown in equation 9 below. The Galerkin formulation of elastodynamics

below introduces the function Uh(x,t), the trial function, which is the approximation of the true

solution U(x,t).

Equation 9. Galerkin formulation of 1D elastic bar BVP

The formulation for uh(x,t) and wh(x,t) are shown below in eq 10 where NEN is the number of

element nodes. The 1D elastic BVP requires the use of 2 node elements and therefore each

element has an N1 and N2 shape function. The 2 node elements used classify Uh as a first

order approximation of the solution. It is first order because the shape functions, N1 and N2 for

each element, are linear functions.

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Equation 10. Finite element approximation of trial and weight functions in space and time

Utilizing the finite element approximations stated in equation 10, the two terms from the galerkin

formulation in equation 9 above are shown to be the mass (M) and stiffness (K) matrices as

shown below where x double dot is acceleration and x is displacement.

Equation 11. Mass and Stiffness matrices

Utilizing 2 node elements, the individual element mass and stiffness matrices are shown below.

The problem requires the calculation of two different types of mass matrices, a lumped mass

and consistent mass. The lumped mass is a diagonal matrix and the consistent mass matrix is

not. The definition of each is shown below:

Equation 12. Element Mass and Stiffness matrices

Equation 11 and the strong form above lead to the semi discrete equation

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Equation 13. Semi-Discrete equation of 1D elastic bar BVP

Where a(t) is the acceleration at time t and d(t) is the displacement at time t. The spatial

component of the finite element approximation is taken care of in the stiffness matrix K. The full

discrete equation is achieved and shown below in equation 14.

Equation 14. Full discrete equation of 1D elastic bar BVP

The dn, vn and an are the approximations of the true or exact values: d(tn), v(tn) and a(tn). From

the discrete equation, the Newmark method of time integration is utilized to calculate the

approximations of the displacement, velocity and acceleration at each node over a specified

time period. The stress as functions of space and time is calculated using the stress strain

relationship from equation 5 above while strain is the derivative of displacement with respect to

x. The Newmark time integration method is summarized below for both implicit and explicit

solutions, the explicit solution is when β = 0 where n+1 is the time step after the nth time step

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Equation 15. Newmark Time integration method

The implicit and explicit methods are both developed via the Newmark method however it’s

important to explain the significance of each. The explicit method calculates a state of the

system at a later time from the state of the system at the current time and the implicit method

takes a weighted average of the current and predicted future state of a system. The explicit

method requires generally smaller and greater number of time steps to approximate the true

solution in order to minimize the error while the implicit method requires extra computations and

matrix inversions. The error associated with both methods of approximation will be discussed

further in the discussion section of this paper.

To begin the cascade of time iterations, begin at time t=0 and calculated a0. D0 and v0 are

known, to calculate a0 the time t=0 discrete equation is set up, subtract the Kd0 term and

multiply it by the inverse of the mass matrix. Utilizing the equations in equation 15 above, begin

calculating d(tn+1), v(tn+1) and a(tn+1) until the end of the simulation has been reached.

The rationale for grouping the central difference method with the consistent mass and the

average acceleration with the lumped mass matrix is to minimize error. The theory of matched

methods suggests that transient integrators and mass matrices shall be matched so that the

induced period errors from the integration method and mass matrix chosen tend to cancel. The

trapezoidal rule of temporal integration tends to increase the simulation time periods and the

converse is true of the central difference method. Similarly, the lumped mass matrix has a

tendency to increase the simulation time period and the opposite is true of the consistent mass

matrix. This is proven via the exact solution of the scalar version of the semi discrete equation of

the 1D elastic bar BVP. The solution is seen below:

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Equation 16. Ratio of temporal and spatial integration frequency to exact frequency

Where ∆t is the time step of the temporal integration, β = Newmark coefficient, h is element

length, ω is the exact frequency response and is the frequency produced by time integration

in conjunction with the element spatial discretization. The equation 16 above is graphically

summarized in the two figures below that represent the rationale for the matching method

chosen.

Figure 2. Matched method rationale

The figure on the left represents the frequency ratio (ideal is 1) with curve 1 representing the

consistent mass and curve 2 is the lumped mass. The figure on the right shows the ratio of the

time step to simulation period vs algorithmic damping ratio.

The match up of the central difference method and the lumped mass matrix is a unique case in

which the period errors from the spatial discrete system and the particular integrator are

canceled perfectly and the exact solution is achieved. With β =r=0 and the time step = h/c, =

ω and this property is called superconvergence.

Stability of the Newmark integration method for the 1D elastic rod BVP is summarized below

where β and γ are the Newmark integration constants.

c c

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Equation 17. Stability analysis of Newmark Time Integration

The ω is the square root of the maximum eigenvalue of the K and M matrix. Ultimately, for 1 D

wave equation, the critical time step for conditionally stable time integration methods (central

difference method utilizing the consistent mass) is the time it takes for the wave to travel

through one element. The critical time step is reduced to the equation below:

Equation 18. Critical Time step for central difference method

Where h is the length of each element and ρ is the material density. If any time step above the

critical time step is employed for a conditionally stable approximation, the model will not

converge and the approximation error goes to infinity. More details on convergence and the

derivation of the stability conditions are in the discussion section including an example of a

displacement approximation that does not converge. It’s important to note that because of the

physical significance of the critical time step, reducing the time step while holding the mesh

length fixed can only worsen the results and for this investigation, unless otherwise specified,

the critical time step will be used.

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The accuracy of the approximations will be shown a number of different ways. The accuracy of

both the displacement and stress approximations from the average acceleration and consistent

mass method will be calculated using the L2 and H1 error norms, respectively. For different time

steps and element sizes visual representations of the error will be provided to demonstrate the

trends. Below are the equations used to calculate the L2 and H1 error norms:

Equation 19. L2 error norm

Equation 20. H1 error norm

Again, u is the exact solution and uh is the approximate solution and thus the difference

between the approximate and exact solutions over the length of the bar is of interest. It’s

necessary to calculate an error norm at individual time steps and averaging them over the

length of the rod, so at any discrete time step the difference between the two solutions is of

interest.

A log-log plot of element size versus error norm will be provided to quantify the accuracy of the

finite element approximations and the convergence. The slope of the log-log line is the rate of

convergence, α, for the respective approximation model from the equation below

Equation 21. Rate of Convergence

Where h is the element length and c is a constant.

The discussion section will explain how the equations presented in this section demonstrate

how element size and time step size affect both stability and accuracy of the finite element

approximate solution.

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4 RESULTS

Before displaying the results, the notations in the plot legends L, L/4, L/2 etc represents

positions along the bar. L/2 represents the mid point, L represents the end of the rod impacting

the wall and 0 represents the free end of the bar.

The figures 3 and 4 below are the plots of displacement and stress, at various points along the

bar, over the simulation time period utilizing the consistent mass and average acceleration time

integration.

Figure 3. Approximate solution of Displacement vs Time

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Figure 4. Approximate Solution of Stress vs Time

The bar was divided up into twenty 2 node elements of the same length. The legend indicates

which line corresponds to the various locations along the bar. An important note, the stress is

calculated on each element whereas the displacements are calculated at each node.

Figures 5 and 6 below are the plots of the exact displacement and stress over the simulation

time period utilizing the lumped mass and central difference time integration. Super

convergence is defined as when a finite element approximation yields the exact solution as the

lumped mass and central difference method does.

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Figure 5. Exact Solution of Displacement vs Time

Figure 6. Exact Solution of Stress vs time

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Identical to the approximate solution, the bar was divided up into twenty 2 node elements of the

same length. The legend indicates which line corresponds to the various locations along the

bar. An important note, the stress is calculated on each element whereas the displacements are

calculated at each node.

The figure below superimposes the real solution and approximate solutions of displacement at 2

different points along the bar for a visual comparison.

Figure 7. Comparison of displacement approximations to exact solution

The figure below superimposes the real solution and approximate solutions of stress at 2

different points along the bar for a visual comparison.

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Figure 8. Comparison of stress approximations to exact solution

The graph below shows how the finite element approximate gets closer to the exact solution as

element size is increased. The graph below displays the exact displacement at the midpoint

(Length/2) of the bar with the approximate values utilizing 20 and 100 element discretizations.

Figure 9. Effect of element size on approximation

The approximations are the result of the implicit method of integration and the consistent mass

matrix and the exact solutions are a result of the lumped mass and central difference method.

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The approximate solutions are compared to the exact solutions of the same element size for

reasons covered in the discussion section.

The graph below shows how the error of the approximate solution is improved with smaller time

steps.

Figure 10. Effect of time step size on approximation

Each curve is representative at the stress at the midpoint of the bar (L/2) utilizing a 100 element

discretization, the different time steps used for the approximate solutions are indicated in the

legend where h is element length

A more detailed discussion on the results presented, including but not limited to the accuracy of

the approximations and stability of the models will be discussed in the next section.

5 DISCUSSION

The approximate solutions of displacement in figure 3 are reasonable shapes when considering

what physically happens to the bar during impact into a rigid wall. In figure 3, the displacement

at the wall end (x=L), stays zero, consistent with the boundary conditions, and the displacement

has a peak at the free end (x=0) halfway through the time simulation. This makes sense

because the free end continues to displace until the wave induced by the impact of the opposite

end of the bar and the wall reaches the free end and sends the first node back to its nominal

position. Applying the wave explanation to the other nodes, the graph indicates that the nodes

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closer to the wall end encounter the wave from impact sooner because it has less distance to

travel which makes sense.

The approximate solutions of the stresses in figure 4 exhibits the overall behavior one would

expect from a physical interpretation however there are some areas on the curves that do not

correlate to the physical reaction. Stress is equal to strain times the elastic modulus, a constant

in this case, and the stress remains zero for elements until they change length. Again, the

boundary condition, at L, the stress remains zero however for an element at the free end, the

element doesn’t crunch until the wave has time to propagate through the bar. Once the bar

returns to its nominal position that same element returns to its original length and the stress

goes back to zero as seen on the graph. The overshoot on the max stresses at various points

along the bar and at the end of the simulation when the bar returns to its nominal length the

stress crosses the x-axis similar to an over damped response are not representative of the

model. In this simulation no damping is assumed and thus these overshoots are clearly errors in

the approximation.

The exact solutions in figures 5 and 6, as stated above, are the results from the lumped mass

and central difference integration method. From the methods section this is the result of

superconvergence. The graphs are identical to the expected response of the bar. The

assumptions made and boundary conditions imposed, the problem reduces to a series of linear

elastic springs (represented by the elements) in which they compress and then release back to

their nominal state. The exact solution shows that the stress reaches the same exact maximum

value (-1) for all elements because they all compress to the same length eventually and then

return to their nominal length (constant material properties) e.g. stress goes back to zero.

As a reminder, for the purposes of this paper the lumped mass with central difference temporal

integration method will be referred to as the exact solution and the consistent mass with the

average acceleration method as the approximate solution.

The comparisons of displacement and stress to the exact solution at various points are included

simply to show on the same graphs, how the different methods’ results look while superimposed

upon each other. Later in this section, the accuracy of the model with respect to time step length

and element size will be discussed.

Stability of a finite element model is best described as the ability of the model to converge and

produce reasonable results. The reason this is important because stability depends on the time

step size and element dimension selected, if too large a time step is used or the elements are

too large, the resolution will be to coarse and the correct response will not be captured. Similar

to the Nyquist theorem for data sampling rates, the theorem states that a signal must be

sampled at a minimum of 2x the frequency at which the signal being measured is oscillating. If

the temperature profile of an engine block of a drag racing car is of interest and the temperature

is recorded every 5 minutes, the resolution of the curve will not provide the detail of interest. In 5

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minutes the car will go from idle to maximum temperature and then the engine will then cool off

presumable back to the idle temperature and the data will be of no value. If the rotor

temperature of a steady state generator was of interest, recording the temperature ever 5

minutes could be appropriate as the temperature will not vary much over a 5 minute period.

Conversely, it would be a waste of time and data storage to measure the generator rotor

temperature every half second frequency during stead operation however the every half second

would be reasonable for obtaining a temperature profile of a drag car engine during a race. The

finite element models are the same way, small elements or short time steps between each data

point may not be necessary however too large of elements or too large a time step might not

give a reasonable response. Finite element models that utilize extremely small time steps and

element sizes are costly and time consuming and in some cases necessary but in other cases

not accurate and thus the appropriate time steps and element sizes must be chosen for the

specific model.

With the superconvergent condition, it’s important to note that the accuracy of the lumped mass

and central difference method is not improved with varying element size or smaller time steps.

Theoretically the accuracy of the values of stress and displacement at each node are

independent of element size however with minute numerical calculations propagated through

the time integration, different element sizes produce fractionally small displacement and stress

magnitudes. This is why figure 9 utilizes two different element sizes for comparison to the

approximation method. For reasons stated in the methods section, the critical time step is the

only time step that produces the exact results at each node and that is why the effect of the time

step was not studied for the lumped mass and central difference approximation method.

The stability of the superconvergent condition is however affected by the time step chosen. The

dependence on time step size for both methods is derived from the eigenvalue problem of the

semi-discrete equation presented in the methods section. From the modal equation the

amplification matrix, the ratio of a time step value to the previous time step value, requires

conditions that prevent the amplification matrix becoming larger and larger at later time steps.

The spectral radius is defined as the maximum eigenvalue of the amplification matrix and it

must meet two conditions, the maximum value of the spectral radii must be less than or equal to

1 (less than 1 if eigenvalues are not distinct) and the eigenvalues of the amplification matrix of

multiplicity greater than one are strictly less than one in modulus. A violation of these two

conditions produces rapid or slow divergent instabilities as seen in the figure below. The figure

below is an example of the same 20 element approximation however the time step is too large

for the model to converge.

Form the methods section and equation 17, the central difference method was stable for all time

steps below the critical time step and the average acceleration method with consistent mass

was stable for any time step chosen.

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Figure 11. Unstable finite element model results

The only difference between figure 11 and figure 5, the exact displacement, is that the time step

used was 1.2 times the critical time step which lead to an unstable solution.

Figure xx in the methods sections summarizes the stability conditions for the specific problem of

the 1D elastic bar. From that figure one can see that the critical time step is the element length

divided by the speed of sound in the given material. This makes physical sense, as discussed

before this is the time it takes for the wave to propagate through one element. As evidenced by

figure 11 above, if this time step is 1.2 times the critical time step, very small increase, the

proper response cannot be captured. Additionally, it becomes clear that the time step is

dependent on element size as well.

The implicit time integration method, average acceleration, with a consistent mass matrix is

stable for any time step chosen. This is because from equation 17 in the methods section, with

beta = ¼ and gamma = ½, the critical sampling frequency is zero. Although any time step can

be chosen, it’s important to note that the error of the approximation is directly related to the time

step squared. The model will converge for any time step however the accuracy is directly

proportional to time step squared so it’s important to know how accurate the approximations

must be for a given application. Figure 10 from the results points to the same conclusion. The

figure shows a 100 element discretization using the average acceleration and consistent mass

matrix with smaller and smaller multiple time steps of the critical time step, h/c, and visually the

conclusion can be drawn that the smallest time step h/4c is the closest to the exact solution

using a 100 element discretization.

Outside the scope of this study, the high frequency oscillations seen in the approximate

solutions in figure 12 can be eliminated using the consistent mass matrix coupled with different

beta and gamma values. For gamma values above ½ there is a beta value, [(gamma + ½)^2]/4,

that optimizes the damping of the higher frequency oscillations. The figure below shows

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compares the exact solution to the non damped approximation and the optimally damped

approximation with gamma = 0.6 and beta = 0.3025 (black dotted line).

Figure 12. Damping of high frequency oscillations in approximate solution

For the case of the 1D elastic bar, the exact solution is provide via the superconvergent

condition and thus the error for the approximated displacements and stresses can be calculated

from the average acceleration method versus element size. From figure 9 in the results, it’s

clear that time step being equal, the smaller the element size the better the approximation.

Now that a visual presentation of the error in finite element approximation has been provided,

the error norms of displacement and stress are quantified via the equations 19 and 20

presented at the end of the methods section. The L2 error norm calculated for displacement

error is graphed versus different element lengths below. The plot is a log-log plot where the

natural log of the element dimension and the error norm are displayed.

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Figure 13. L2 Error norm of Displacement

T2 represent a specific time step, the error norm was calculated on a single time step for

various numbers of elements. The slope of the graph, 2, is equal to the rate of convergence and

of the displacement error which is in line with the theoretical value assuming the exact solution

is infinitely continuous e.g. a sinusoidal function. Most importantly, the trend is clear that the

error is reduced with a smaller element size e.g. finer mesh.

Similar for the error on stress, the figure below is the H1 error norm calculated for models with

different element sizes.

Figure 14. H1 Error Norm of Stress

T2 represent a specific time step, the error norm was calculated on a single time step for

various numbers of elements. The slope of the graph, 1, is equal to the rate of convergence and

of the stress error which is in line with the theoretical value assuming the exact solution is

infinitely continuous e.g. a sinusoidal function. Again, the trend is clear that the error is reduced

with a smaller element size.

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6 SUMMARY & CONCLUSIONS

The two methods analyzed of approximating the response of the elastic bar problem provide

both exact and reasonable responses when the parameters chosen are best understood. The

average acceleration method utilized with the consistent mass provided a reasonable

approximation regardless of the time step chosen and was better approximated with smaller

time steps and smaller element sizes. The central difference method coupled with the lumped

mass provided the exact solution at each node only for time steps equal to the critical time step.

Theoretically the element size, for the central difference and lumped mass method should not

have affected the results at a given node however due to minute computational errors

compounded through the time integrations, the element size did change the magnitude of the

displacements and stresses by fractions of percentage points.

For practical purposes and not discussed in detail, the understanding of the limitations of a finite

element model are extremely important when taking into consideration time and cost

constraints. As evidenced by the results of this investigation, one approximation provided better

results with smaller time steps and element sizes e.g. more expensive and timely however the

other approximation distanced itself further from the exact solution with smaller time steps and

similarly no accuracy was gained with smaller element sizes.

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7 REFERENCES

1) Hughes, J. R., The Finite Element Method: Linear Static and Dynamic Finite Element

Analysis, Dover Publications, Inc, Mineola, New York.

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APPENDIX A. MATLAB CODE

%CEE 235B Final Project

%John Wick % clear all; close all; clc

%number of elements n=10;

%Initial Conditions at all nodes @ time t=0 %initial Velocity = 1 %initial Displacement = 0 %initial Acceleration = 0

%known Material Properties/Constants (dimensionless) E=100; %Bar Elastic Modulus L=10; %Bar Length A=1; %Bar Cross Sectional Area rho=0.01; %Bar Density c=sqrt(E/rho); %Bar Material Speed of Sound L_e=L/n; %length of element

%number of degrees of freedom ndof=2; %1 dimensional problem, 2 nodes

%stiffness matrix for 2 node element 1D elasticity (slide 58) K = zeros(n+1,n+1);

for i=1:n K(i,i)=2*E*n/L; K(i,i+1)=-E*n/L; K(i+1,i)=-E*n/L; K(i+1,i+1)=2*E*n/L; end %eliminate the last column and row -- last node displacement = 0 K(1,1)=E*(n/L); K(n+1,n+1)=E*(n/L); K(n+1,:)=[]; K(:,n+1)=[]; % display('Global stiffness matrix,');K

%Construct global consistent mass matrix M_c = zeros(n+1,n+1); for ii=1:n M_c(ii,ii)=(2/3);

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M_c(ii,ii+1)=1/6; M_c(ii+1,ii)=1/6;

end M_c(1,1)=1/3; M_c(n+1,n+1)=1/3; M_c=M_c*(rho*L_e);

%Lumped Mass Matrix for 2 node elements 1D elasticity

M_l = zeros(n+1,n+1);

% for j=1:n % M_l(j,j)=M_c(j,j); % end % M_l(n+1,n+1)=M_c(n+1,n+1); M_c_sum=sum(M_c,2);

%place M_c_sum elements into diagonals of M_l for ROW SUM Lumped Mass for u=1:n+1; M_l(u,u)=M_c_sum(u,1); end

%Choose lumped mass or consistent mass M=M_c;

%Define Beta and Gamma based on Mass Matrix Chosen Above if M==M_c B=1/4; gamma=1/2; %Simulation Time Constants D_t=(2*L_e)/(c); %Critical time step for trapezoidal rule with

consistent mass (unconditionally stable) T_total=0.2; %duration of the whole simulation t=T_total/D_t; %total number of time steps else if M==M_l B=0; gamma=1/2; %Simulation Time Constants D_t=L_e/c; %Critical time step for central difference with lumped mass

(critical time step slide 59) T_total=0.2; %duration of the whole simulation t=T_total/D_t; %total number of time steps end end

%eliminate the last column and row -- last node displacement = 0 (EBCs) M(n+1,:)=[]; M(:,n+1)=[];

% display('Global Mass matrix,');M

%Initial displacement Vector -- eliminate the last node displacement because

its 0

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d_initial=zeros(n,1);

%Initial velocity Vector -- eliminate the last node velocity because its 0 v_initial=zeros(n,1); for y=1:n v_initial(y,1)=v_initial(y)+1; end

%Call displacement, velocity and acceleration matrices -- Pre Allocation displacement=zeros(t+1,n+1); velocity=zeros(t+1,n+1); acceleration=zeros(t+1,n+1);

%Initial acceleration vector @ t=0 a=M\-K*d_initial;

%Place iniital time steps into Displacement, Velocity & Acceleration matrices d=d_initial; v=v_initial; displacement(1,1:n)=d_initial; velocity(1,1:n)=v_initial; acceleration(1,1:n)=a;

for k=1:t %predictors d_p=d+D_t*v+0.5*((1-2*B)*((D_t)^2)*a);

v_p=v+((1-gamma)*(D_t)*a);

% M star and F star M_s=M+(B*((D_t)^2)*K); F_s=-K*d_p;

%acceleration vector a_n+1 a=M_s\F_s;

%Correctors d=d_p+(B*(D_t^2)*a); v=v_p+(gamma*(D_t)*a);

%assembly of looped displacement, velocity and acceleration matrices displacement(k+1,1:n)=d; velocity(k+1,1:n)=v; acceleration(k+1,1:n)=a; end

%Assembly of the Stress matrix stress=zeros(t,n+1);

for r=1:t

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for rr=1:n stress(r,rr)=(E*(displacement(r,rr+1)-displacement(r,rr)))/L_e; end end dx=zeros(t,n+1); for r=1:t for rr=1:n dx(r,rr)=((displacement(r,rr+1)-displacement(r,rr)))/L_e; end end

%Exact solution M=M_l; M(n+1,:)=[]; M(:,n+1)=[]; B=0; gamma=1/2; %Simulation Time Constants D_t=L_e/(c); %Critical time step for central difference with lumped

mass (critical time step slide 59) T_total=0.2; %duration of the whole simulation t=T_total/D_t; %total number of time steps

%Call displacement, velocity and acceleration matrices -- Pre Allocation displacement_exact=zeros(t+1,n+1); velocity_exact=zeros(t+1,n+1); acceleration_exact=zeros(t+1,n+1);

%Initial acceleration vector @ t=0 a=M\-K*d_initial;

%Place iniital time steps into Displacement, Velocity & Acceleration matrices d=d_initial; v=v_initial; displacement_exact(1,1:n)=d_initial; velocity_exact(1,1:n)=v_initial; acceleration_exact(1,1:n)=a;

for kk=1:t %predictors d_p=d+D_t*v+0.5*((1-2*B)*((D_t)^2)*a);

v_p=v+((1-gamma)*(D_t)*a);

% M star and F star M_s=M+(B*((D_t)^2)*K); F_s=-K*d_p;

%acceleration vector a_n+1 a=M_s\F_s;

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%Correctors d=d_p+(B*(D_t^2)*a); v=v_p+(gamma*(D_t)*a);

%assembly of looped displacement, velocity and acceleration matrices displacement_exact(kk+1,1:n)=d; velocity_exact(kk+1,1:n)=v; acceleration_exact(kk+1,1:n)=a; end

% figure(2); % plot(displacement_exact) % format longG % g=max(displacement_exact(:,n/2))

%Assembly of the Stress matrix

stress_exact=zeros(t,n+1);

for r=1:t for rr=1:n stress_exact(r,rr)=(E*(displacement_exact(r,rr+1)-

displacement_exact(r,rr)))/L_e; end end

dx_exact=zeros(t,n+1); for r=1:t for rr=1:n dx_exact(r,rr)=(displacement_exact(r,rr+1)-displacement_exact(r,rr))/L_e; end end

%error

%subtract each displacement and square it (U - Uh)^2

error=zeros(t+1,n+1);

for y=1:t+1 for yy=1:n+1 error(y,yy)=(displacement_exact(y,yy)-displacement(y,yy))^2; end end

%integrate the curve at each displacement (trapezoid rule) and make an error

vector that has %an error at each time step

E_0=zeros(t,1); error_pre_sum=zeros(t+1,n+1);

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for o=1:t for oo=1:n error_pre_sum(o,oo)=(error(o,oo)+error(o,oo+1))*(n/(L*2)); end end

%need to sum the row elements of each row and place them in E_0 giving you %error norm at each time step for mm=1:t E_0(mm,1)=sqrt(sum(error_pre_sum(mm,1:n))); end

% figure(3); % plot(E_0)

%error of derivative ie stress!

%subtract each displacement and square it (U - Uh)^2

error_dx=zeros(t,n+1);

for y=1:t for yy=1:n+1 error_dx(y,yy)=((dx_exact(y,yy)-dx(y,yy))^2); end end

%integrate the curve at each displacement (trapezoid rule) and make an error

vector that has %an error at each time step

E_0_dx=zeros(t,1); error_dx_pre_sum=zeros(t+1,n+1); for o=1:t for oo=1:n error_dx_pre_sum(o,oo)=(error_dx(o,oo)+error_dx(o,oo+1))*(n/(L*2)); end end

% need to sum the row elements of each row and place them in E_0 giving you % error norm at each time step for mm=1:t E_0_dx(mm,1)=sqrt(sum(error_dx_pre_sum(mm,1:n))); end

% figure(4); % plot(stress)

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CEE 235B Final Project_John_Wick