ce 257 discrete methods of structural analysis

CE 257 Discrete Methods of Structural Analysis

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This lecture material is not for sharing/distribution outside of CE257 and CE297 class.

Last meeting

• Eigenvalue Problems

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For this meeting

• Time-dependent Problems

Reference for this lecture:

J. N. Reddy, An Introduction to the Finite Element Method (Chapter 6)

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CE28 Analytical and Computational Methods in CEIII

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Time-dependent Problems: Introduction

Introduction

• We develop the finite element models of one-dimensional time-dependent problems anddescribe time approximation schemes to convertODEs in time to algebraic equations.

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Introduction

• F.E. models of time-dependent problems can bedeveloped in two alternative ways: (a) coupledformulation in which the time 𝑡 is treated as anadditional coordinate along with the spatialcoordinate 𝑥 and (b) decoupled formulation wheretime and spatial variations are assumed to beseparable. Thus, the approximation in the twoformulation takes the form

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Introduction

• where 𝜓𝑗𝑒 𝑥, 𝑡 are time-space (two dimensional)

interpolation functions and ො𝑢𝑗 are the nodal values

that are independent of 𝑥 and 𝑡,

• where 𝜓𝑗𝑒 𝑥 are the usual one-dimensional

interpolation functions in spatial coordinate 𝑥 onlyand the nodal values 𝑢𝑗

𝑒 𝑡 are functions of time 𝑡

only.

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Introduction

• In this section, we consider the space-timedecoupled formulation only.

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Introduction

• The space-time decoupled finite elementformulation of time-dependent problems involvestwo steps:

• 1. Spatial approximation - where the solution 𝑢 ofthe equation under consideration is approximatedby the second expression and the spatial finiteelement model of the equation is developed usingthe procedures of static or steady-state problemswhile carrying all time-dependent terms in theformulation. This step results in a set of ODEs intime for the nodal variables 𝑢𝑗

𝑒 𝑡 of the element.

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Introduction

• 2. Temporal approximation - where the system ofODEs are further approximated in time, oftenusing finite difference formulae for the timederivatives. This step allows conversion of thesystem of ODEs into a set of algebraic equationsamong 𝑢𝑗

𝑒 at time 𝑡𝑠+1 = 𝑠 + 1 Δ𝑡, where Δ𝑡 is

the time increment and s is a nonnegative integer.

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Introduction

• All time approximation schemes seek to find 𝑢𝑗 at

time 𝑡𝑠+1 using the known values of 𝑢𝑗 from

previous times:

• Thus, at the end of the two-stage approximation,we have a continuous spatial solution at discreteintervals of time

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Introduction

• Note that the approximate solution has the sameform as that in the separation-of-variablestechnique used to solve the boundary value andinitial value problems. By taking nodal values to befunctions of time, we see that the spatial points inan element take on different values for differenttimes

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Introduction

• We study the details of the two steps byconsidering a model DE that contains both thesecond- and fourth-order spatial derivatives andfirst- and second-order time derivatives.

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Introduction

• The above equation is subject to appropriateboundary and initial conditions. The boundaryconditions are of the form

• at 𝑥 = 0, 𝐿.

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Introduction

• The initial conditions involve specifying

• where ሶ𝑢 ≡𝜕𝑢

𝜕𝑡.

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Introduction

The above D.E. describes the following physicalproblems:

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Semidiscrete F.E. models


• The semidiscrete formulation involvesapproximation of the spatial variation of thedependent variable. The formulation followsessentially the same steps as described in theprevious lessons.

• The first step involves the construction of theweak form of the equation over a typical element.

• In the second step, we develop the F.E. model byseeking approximation of the form in thedecoupled formulation.

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Introduction

• Recall: The boundary conditions are of the form

• at 𝑥 = 0, 𝐿.

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• Next, we assume that 𝑢 is interpolated by anexpression of the decoupled formulation whichimplies that, at any arbitrary fixed time 𝑡 > 0, thefunction 𝑢 can be approximated by a linearcombination of 𝜓𝑗

𝑒 and 𝑢𝑗𝑒 𝑡 , with 𝑢𝑗

𝑒 𝑡 being

the value of 𝑢 at the 𝑗th node of the element Ω𝑒.

• We obtain the F.E. solution in the form

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• where (𝑢𝑗𝑠)𝑒 is the value of 𝑢(𝑥, 𝑡) at time 𝑡 =

𝑡𝑠 and node 𝑗 of the element Ω𝑒.

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• The above equation is a hyperbolic equation, andit contains the parabolic equation as a special case(set [𝑀2] = [0]).

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Semidiscrete F.E. models:Parabolic Equation

Parabolic Equations

• Time Approximation

• Consider the following D.E.:

• In the finite difference of the D.E., we replace thederivatives with their finite differenceapproximation.

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Parabolic Equations

• The equation can be expressed as

• When 𝛼 = 0:

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Forward difference approximation

Parabolic Equations

• The equation can be expressed as

• When 𝛼 = 1:

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Backward difference approximation

Parabolic Equations

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Parabolic Equations

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• Approximations to the D.E.:

Parabolic Equations

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• Time marching equation:

Parabolic Equations

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Time approximation scheme using finite element method:

• Approximation of solution:

Parabolic Equations

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• Time approximation scheme using finite element method:

• Approximation of solution:

• Approximation of 𝑓(𝑡):

Parabolic Equations

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• Time approximation scheme using finite element method:

• If 𝑢𝑠 is known:

Parabolic Equations

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• Stable and Conditionally Stable Schemes

• When the error remains bounded for any time step, the scheme is stable.

• If the error remains bounded only when the time step remains below certain value, the scheme is said to be conditionally stable.

Parabolic Equations

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• Stable and Conditionally Stable Schemes

• For different values of 𝛼 , the following are the well-known time approximation schemes:

Fully discretized finite element equations

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• Matrix equation:

• Subject to initial condition:

• We obtain:

Consistency, Accuracy, Stability

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• The numerical scheme is said to be consistent if the round-off and truncation errors go to zero as Δ𝑡 → 0 .

• Accuracy is a measure of closeness between the approximate solution and exact solution.

• Stability of a solution is a measure of the boundedness of the approximate solution with time

• A time-approximation scheme is said to be convergent if, for fixed 𝑡𝑠, the numerical value 𝑢 𝑠 converges to its true value {𝑢(𝑡𝑠)} as Δ𝑡 → 0 .


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Semidiscrete F.E. models:Hyperbolic Equation

Hyperbolic Equations

• Time Approximation

• Consider matrix equations of the form

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• Such equations arise in structural dynamics, where[𝑀] denotes mass matrix, [𝐶] the damping matrix,and [𝐾] the stiffness matrix. The damping matrix[𝐶] is often taken to be a linear combination of themass and stiffness matrices, [𝐶] = 𝛽1[𝑀] +𝛽2[𝐾], where 𝛽1 and 𝛽2 are determined fromphysical experiments.

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Related Problems:

• Axial Motion of Bars

• Transverse Motion of Euler-Bernoulli Beams

• Transverse Motion of Timoshenko Beams

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• There are several numerical methods available toapproximate the second-order time derivativesand convert differential equations to algebraicequations.

• Newmark family

• Wilson method

• Houbolt method

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• Newmark's Scheme

• In the Newmark Method, the function and its firsttime derivative are approximated according to

where

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• The 𝛼 and 𝛾(= 2𝛽) are parameters thatdetermine the stability and accuracy of thescheme. Equations can be viewed as Taylor's seriesexpansion of 𝑢 and ሶ𝑢.

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• The following are special cases:

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• For all schemes in which 𝛾 < 𝛼 and 𝛼 ≥1

2, the

stability requirement is

• where 𝜔max is the maximum natural frequency ofthe system from (if without [𝐶])

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Fully Discretized Finite Element Equations

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• Initial conditions: 𝑢 0 , ሶ𝑢 0, ሷ𝑢 0

• Calculate ሷ𝑢 0 from:

• Update vectors:

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Mass Lumping

Mass Lumping

• Recall from the time approximation of theparabolic equations that use of the forwarddifference scheme (i.e., 𝛼 = 0 ) results in thefollowing time marching scheme

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Fully discretized finite element equations

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• Matrix equation:

• Subject to initial condition:

• We obtain:

Mass Lumping

• The matrix [𝑀𝑒] derived from the weighted-integral formulations of the governing equation iscalled the consistent mass matrix, and it issymmetric positive-definite and non-diagonal.Solution of the global equations requires inversionof the assembled mass matrix.

• If the mass matrix is diagonal, then the assembledequations can be solved directly (i.e. withoutinverting a matrix)

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Mass Lumping• There are several ways of constructing diagonal

mass matrices, also known as lumped massmatrices.

• The use a lumped mass matrix in transientanalyses can save computational time in two ways:

• First, for the forward difference schemes, lumpedmass matrices result in explicit algebraic equationsnot requiring matrix inversions.

• Second, the critical time step required forconditionally stable schemes is larger, and henceless computational time is required when lumpedmass matrices are use.

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Mass Lumping

Row-Sum Lumping

• The sum of the elements of each row of theconsistent mass matrix is used as the diagonalelement

• where the property σ𝑗=1𝑛 𝜓𝑗

𝑒 = 1 of the

interpolation functions is used.

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Mass Lumping

Row-Sum Lumping

• When 𝜌 is constant:

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Mass Lumping

Row-Sum Lumping

• Consistent mass matrices

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Mass Lumping

Proportional Lumping

• Here the diagonal elements of the lumped massmatrix are computed to be proportional to thediagonal elements of the consistent mass matrixwhile conserving the total mass of the element:

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Mass Lumping

Proportional Lumping

• For constant 𝜌, proportional lumping gives thesame lumped mass matrices as those obtained inthe row-sum technique for the Lagrange linearand quadratic elements.

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Homework

• Derive the fully discretized finite element equation

using Newmark beta (𝛾 =1

2, 𝛽 =

1

4):

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1

2

1

2

44242 ++ ++

++

+

=

+

+ nnnnn

tttttfuMuMCuMCuMCK


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Finite Element Method

References:

1. J. Fish, T. Belytschko, A First Course in Finite Elements

2. J. N. Reddy, An Introduction to the Finite Element Method

3. O.C. Zienkiewicz, R.L. Taylor, J.Z. Zhu, The Finite Element Method, Its Basis & Fundamentals

4. K-J. Bathe, Finite Element Procedures

5. K. Leet, C-M., Uang, A.M. Gilbert, Fundamentals of Structural Analysis

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This lecture

• Time-dependent Problems

Reference for this lecture:

J. N. Reddy, An Introduction to the Finite Element Method (Chapter 6)

67CE 257

This lecture material is not for sharing/distribution outside of CE257 and CE297 class.


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ce 257 discrete methods of structural analysis

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