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1. ES 128 Computational Solid and Structural Mechanics

2. 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

3. 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

4. Introducing yourself

5. Introduction • What do l learn in ES128? • Why is ES128 important for me? • How is ES128 organized?

6. 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.

7. 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

8. 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.

9. 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.

10. 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

11. How is ES128 organized? Spring 2010: Tue – Thu –1:00 pm - 2:30 pm Lab session: weekly – to be organized Office hours: KatiaThu 3 pm -4 pm Bas Wednesday 5 pm– 6 pm Textbook: A First Course in Finite Elements Jacob Fish, Ted Belytschko

12. Lab session When? How?

13. How is ES128 organized? • The grade comes from four components • Each 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)

14. ES 128 Project (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) 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.

15. 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

16. 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

17. Engineering Design Physical Problem Question regarding the problem ...how large are the deformations? ...how much is the heat transfer? Mathematical model Governed by differential equations Assumptions regarding Geometry Kinematics Material law Loading Boundary conditions Etc.

18. Engineering Design The mathematical model is often too complicated to solve by hand. We therefore solve it using a numerical technique - the finite element method. Physical Problem Mathematical model Governed by differential equations Numerical model e.g., finite element model

19. Engineering Design Physical Problem Change physical problem Mathematical Model Improve mathematical model Numerical model Does answer make sense? No! Refine analysis Design improvements Structural optimization YES!

20. Finite element method Preprocessing Step 1 Analysis Step 2 Step 3 Postprocessing

21. Preprocessing The problem domain is subdivided into finite elements

22. 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 Ax= b Element Node

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

24. Comments The numerical solution is only as accurate as the mathematical model 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.

25. We’ll start from…..Trusses

26. Bridge How does the bridge deform under theses applied forces? P2 P1