Reviving Continuum Mechanics: Computation across the

undergraduate curriculum

Michael DenninUC Irvine

Special Thanks to Peter Taborek, Bill Heidbrink and Gary Chanan

Two Cases

This poster addresses two issues in the integration of numerical methods into the undergraduate curriculum. One is the integration into a particular course, in this case Continuum Mechanics. The other is the integration into the overall curriculum. There are related issues for both of these problems.

Continuum Mechanics

• Lack of analytically solvable problems

• Tedious amounts of algebra

• Visualization often a problem

• Modern techniques are dominated by numerical methods

Challenges facing the course

Continuum Mechanics

• Choice of numerical method

• Access to numerical techniques by students

• Time devoted to teaching numerical method

Challenges specific to numerical methods

Continuum Mechanics:Solutions at UCI

• Mathematica required of all students

• Mathematica available on UCI computers

• Integration of technology in the classroom

Curriculum Integration

Mathematical Goals of the Curriculum

Master Calculus andBasic Linear Algebra

First two years:• basic problem solving skills.• solves most “basic” physics problems.• basis of “higher” math.

Learn Mathematica:Review “basic” mathUseful numerical/symbolic package

End of Second Year:• excellent tool for learning “higher” mathematics.• access to greater diversity of physics problems.• introduces modern research tool

Important Distinction

• Numerical methods course: fundamentally a programming course that focuses on the challenges of using numerical methods in research.

• Numerical “packages”: fundamentally a tool for solving problems EITHER symbolically or numerically. Can be used for “research type” numerical work, but most powerful in the classroom when used to teach mathematical methods. Symbolic manipulation is a key pedagogical tool.

Main Advantages of Program

All students move beyond “special case” problems.Gain a tool for problem presentation.

Weaker students: focus on physics not math issues.

Stronger students: exposed to even more challenging problems

All students gain a useful tool for future employment in academics AND industry.

Some Typical “Math Experiences”Apparent Goal Actual Result

Solve a particular ODE/PDE Manipulate the known solutions

Learn abstract vector spaces Work with 2 x 2 matrices

Learn about special functionsManipulate relations without intuition

Packages explicitly recognize this and provide flexibility in manipulating/plotting solutions.

Packages increase the complexity of problems that are tractable. N x N matrices not a problem.

Numerics and plotting options increase the opportunity for intuition development.

Challenges of Implementation

• Choice of numerical method: does the department specify one packages or is the tool left to the individual faculty or student?

• Access to numerical tools – cost to the student?

• Instruction: Is a dedicated course used to teach the package, or are methods learned ad hoc?

UCI Specific Solutions

• Mathematica is taught in the spring of the 2nd year.

• Students have access to Mathematica in a Physics computer lab (campus has a campus license).

• Students are free to use any numerical methods in the upper division courses.

Advantages

• Students focus on analytic problem solving for the first two years.

• Faculty flexibility in upper division courses in choice of problem assignments.

• Faculty know students have at least minimal knowledge of Mathematica.

• Students can use Mathematica for excellent presentation of problem solutions.

Disadvantages

• No numerical work in first year, but many introductory concepts are ideal for simple numerical implementation.

• Some students rely too heavily on Mathematica for “simple” manipulations.

• Too many faculty do not assign numerical problems because “they do not know Mathematica.”

Example Student SolutionŸ4.13 - Show that the stream function and velocity potential which are due to the motion of a circular cylinder of

radius a moving with velocity U parallel to the x-axis are: y = - Ua2 sinHqLr

, f = - Ua2cosHqLr

so the first thing to see about a problem like this is that it can be reduced to a two dimensional problem of a circle moving through a fluid. This assumes that the cylinder is infinite in z. We can immediately say that:

f µ cos HqLbecause of the boundary condition that the normal velocity of the fluid has to match the velocity of the cylinder. This settles us on {=1, which also sets the exponent of r. We can eliminate the r{ terms because they don't go to zero at infinity, leaving us with:

fµcos HqLr

finding the constant by unit analysis and matching the boundary condition that

vr r=a= U

and using that v = Ñf, we get

f= -Ua2 cos HqL

r

As for y, we require that Ñ2y=0. following the same analysis that we did for the sphere and using the same BC's as above, we get

y= -Ua2 sin HqL

r

Summary

• Current state at UCI is anecdotal – we need good research. For example, is it necessary to make plots by hand first to have an understanding of plotting? Can students learned certain mathematical concepts better with symbolic manipulation packages?

• Numerical methods are an integral part of physics research, and they need to be taught. It is useful to combine teaching numerical methods as a “research tool” with using it as a pedagogical aid across the curriculum.

• Useful resources help with faculty buy in – at UCI, Peter Taborek has published a Mathematica Handbook: http://mathematicahandbook.com/