ES 128 Computational Solid and

Structural Mechanics

ES128 - Computational Solid & Structural Mechanics

Introduction to Finite Elements

  Focus on computer analysis using the commercial code ABAQUS

Tuesday-Thursday: 1 pm- 2.30 pm + computer lab

Cruft 309

Introducing ourself

Instructor: Katia Bertoldi

Email: bertoldi@seas.harvard.edu

Office: Pierce 311

Office hours: Thursday 3 pm - 4 pm

TF: Johannes (Bas) Overvelde

Email: overvelde@seas.harvard.edu

Office: Pierce 404

Office hours: Wednesday 5 pm - 6 pm

Introducing yourself

Introduction

• What do l learn in ES128?

• Why is ES128 important for me?

• How is ES128 organized?

What do I learn in ES128?

Prerequisite: ES120 and ES123

You will learn computational techniques for the simulation of a large variety of engineered systems. The application to real engineering applications are stressed throughout.

Introduction to finite element methods for analysis of steady-state and transient problems in solid, structural, fluid mechanics, and heat transfer.

We will model problems involving fluids and solids and learn how to interpret the numerical results.

What do I learn in ES128?

Focus on computer analysis using the finite-element method.

In computer implementation, you will develop simple finite-element analyses using Matlab and the general-purpose program ABAQUS.

You will do a project addressing a significant problem

arising in engineering, biomechanics or earth science

Why is ES128 important?

Many engineering problems can be described in terms of partial differential equations

The finite element method is a numerical approach to solve approximately these partial differential equations

FEM is used by million of scientist worldwide.

Why is ES128 important?

In engineering practice, analysis is largely performed with the use of finite element computer programs (such as ABAQUS, NASTRAN, ANSYS, ADINA, SIMULIA, COMSOL etc…)

These analysis programs are interfaced with computer-aided design ( CAD) programs Catia,g ) p g , SolidWorks, Pro/Engineer, NX, etc.

Why is ES128 important?

Computations are everywhere in engineering problems.

Many problems are resolved with the aid of computers and dedicated programs today. It is important to be able to implement numerical algorithms

How is ES128 organized?

Spring 2010: Tue – Thu –1:00 pm - 2:30 pm

Lab session: weekly – to be organized

Office hours: Katia Thu 3 pm -4 pm Bas Wednesday 5 pm– 6 pm

Textbook: A First Course in Finite Elements Jacob Fish, Ted Belytschko

Lab session

When?

How?

How is ES128 organized?The grade comes from four componentsEach component contributes ¼ to the course grade. • Homework (25%)One assignment week (assigned and due on Thursday). Combination of problem sets and computer exercises

• Project (25%)

• 2 Midterms (25% each)

ES 128 Project

You will work in a group of 2-3 students on a project that• addresses a phenomenon or engineering design

issue, and • involves serious use of FEM.

(Step 1) Each student presents a project proposal (Feb 20 in class)

5 minute presentation. On 2 slides you should explain

What is the project? What are the goals? Use at least one figure.

How does FEM contribute to the project

(Step 2) Katia and Bas will use your proposals to formulate N projects

(Step 3) You form groups of 2-3 students and choose a project

(Step 4) Intermediate report + presentation (April 1 in class)

(Step 5) Final report + presentation (Reading Period)

ES 128 Project

The project contributes 25% of the grade, distributed as follows:

10 % April 1, in class. Intermediate presentation (10 minutes) + report.

15%: Reading period Final 15 minute project presentation. Final project report is due

Some previous project titles

• Diffusion and pattern formation in biology• Analysis of a goat pen gate• Blast loading on sandwich beam structure• Wind turbine blade• Analysis of human femur• Rejection seats• Gas gun

Engineering Design

Physical Problem

Mathematical model

Governed by differential equations

Assumptions regardingGeometryKinematicsMaterial lawLoadingBoundary conditionsEtc.

Question regarding the problem...how large are the deformations? ...how much is the heat transfer?

Engineering Design

Physical Problem

Mathematical model

Governed by differential equations

Numerical model

e.g., finite element model

The mathematical model is often too complicated to solve by hand. We therefore solve it using a numerical technique - the finite element method.

Engineering DesignPhysical Problem

Mathematical Model

Numerical model

Does answer make sense?

Refine analysis

YES!

No!

Improve mathematical model

Design improvementsStructural optimization

Change physical problem

Finite element method

Preprocessing

Analysis

Postprocessing

Step 1

Step 2

Step 3

Preprocessing

The problem domain is subdivided into finite elements

Analysis

Step 1: Element formulation - development of equations for the elements

Step 2: Assembly – from equations of a single elements to equations of the entire system

Step 3: Solution of the equations

A x= b

Element

Node

Postprocessing

Determination of quantities of interest (such us stress and strain and their visualization)

0 0.02 0.04 0.06 0.08 0.10

20

40

60

80

100

120

140

Nominal Strain

No

min

al S

tre

ss [k

Pa

]

Comments

For a well-posed mathematical problem the numerical technique should always, for a reasonable discretization, give a reasonable solution which must converge to the accurate solution as the discretization is refined.

The numerical solution is only as accurate as the mathematical model

We’ll start from…..Trusses

Bridge

P1 P2

How does the bridge deform under theses applied forces?