Final Presentation - compmath.files.wordpress.com · Presentation Overview I Project overview and...

Final Presentation

A. Davey, N. Moniz, T. Spilhaus

UMass Dartmouth

August 16, 2012

A. Davey, N. Moniz, T. Spilhaus (UMass Dartmouth) Final Presentation August 16, 2012 1 / 14

Presentation Overview

I Project overview and parameters.

I Motivation for using a particular method to solve the problem at hand.

I Methodology and description of mathematical procedures.

I Numerical results.

I Future goals and possible applications.


Problem Overview and Description

I Using finite element method we will attempt to model the steady-state heatprofile across a flat plate with different boundary conditions and right handside functions.

I We will model Poisson’s equation −∇ · (κ(x , y)∇u(x , y)) = f (x , y) on a unitsquare [−1, 1]2.

I Boundary conditions will begin with homogeneous Dirichlet set equal to zeroand then vary as the project progesses.

I κ(x , y) = 16 + ε1x + ε2y

I In the beginning, ε1 and ε2 will be drawn at random from a normal Gaussiandistribution with σ = 1

2 .


Advantages and Disadvantages of FEM

I Used in a wide variety of areas tosolve and model military,commercial, and industrialproblems.

I Elements can be individuallycustomized to model objects withcomplex geometries in multipledimensions.

I Computationally intensive if highgrid refinement is used and sourcecode can become quite large.

Figure 1: Finite element simulation of jetturbine engine.


Methodology: Outline

I Derive the weak formulation of our partial differential equation.

I Discretize over space (Ω) creating a two-dimensional mesh of arbitraryrefinement.

I Select shape and weight functions.

I Assemble the linear system.

I Apply boundary conditions (either homogeneous Dirichlet or mixed Dirichletand Neumann) and solve system.


Methodology: Deriving the Weak Formulation

−∇ · (κ(x , y)∇u(x , y)) = f (x , y)

We then multiply both sides by a test function v .

−v∇ · (κ(x , y)∇u(x , y)) = vf (x , y)

We now integrate over the domain Ω.

−∫

Ωv∇ · (κ(x , y)∇u(x , y)) =

∫Ωvf (x , y)

Then integrating by parts will yield the following.∫Ω∇v · κ∇u(x , y)−

∫Γvn · ∇u =

∫Ωvf (x , y)

Lastly, we apply homogeneous Dirichlet boundary conditions, u(x , y) = 0, along theboundary Γ.∫

Ω∇v · κ∇u(x , y) =

∫Ωvf (x , y)


Methodology: Basis Functions and Generating the Mesh

I we find an approximation functionof u to satisfy our equation for allfunctions of v.

uh =∑

j cjφj

I we can now define our test functionas shown below.

v = ciφi , i = 0, · · · ,N

I we also need a mesh onto which wewill map our shape functions.

Figure 2: Example of a generated mesh


Methodology: Basis Functions and Generating the Meshcontinued

I then we define the weak form of the discrete problem and substitute in our uh∫Ω∇v · κ∇

∑j cjφj =

∫Ωvf (x , y)

I we then write this in terms of a linear system

Ac = F

I A and F are then replaced with weighted sums over a set of points on eachcell in there domain and we set these as the approximations:

Akij =

∑q∇φi (xkq ) · ∇φj(xkq )wk

q

F ki =

∑q φi (x

kq )f (xkq )wk

q


Numerical Results

I several different conditions applied

I average solutions

I average kappa compared to average solution


Solutions of each Applied Condition

I

I

I

I

Figure 3: Top-Left: homogeneous Dirichlet conditions, Bottom-Left: Mixed Boundaries,Top-Right: added Gaussian on RHS, Bottom-Right: Sigma changed from 1/2 to 5


Average Solutions of each Applied Condition

I

I

I

I

Figure 4: Top-Left: homogeneous Dirichlet conditions, Bottom-Left: Mixed Boundaries,Top-Right: added Gaussian on RHS, Bottom-Right: Sigma changed from 1/2 to 5


Comparison overlay of average solutions with sigma 5 andsigma 1/2

Figure 5: Overlay of average of 100 solutions with σ = 12

in green and σ = 5 in red andblue.


Convergence of average solution to the average kappasolution


Possible Future Goals

I Modeling more realistic physical systems, such as changing our Kappa to bethe actual heat conductivity of a known material.

I Adding in Time Dependance to the problem being modeled and seeing how itevolves.

I Using FEM to approximate higher dimensional systems

I Modeling Higher Order Methods for higher accuracy results


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