thesis_1995_vt

thesis_1995_vt, "Sensitivity Analyses and Computational Shape Optimization for Incompressible Flows", submitted to the Department of Mathematics at Virginia Tech in May 1995. My thesis adviser was Max Gunzburger.

Abstract: We consider the optimization of a cost functional defined for a fluid flowing through a channel. Parameters control the shape of an obstruction in the flow, and the strength of the flow. The problem is discretized using finite elements. Optimization algorithms are considered which use either finite differences or sensitivities to estimate the gradient of the cost functional. Problems of scaling, local minimization, and cost functional regularization are considered. Methods of improving the efficiency of the algorithm are proposed.

The most interesting thing to me about this thesis was the investigation of the properties of the sensitivities, which estimate how much the flow quantities change when some parameter is changed.

At first, I assumed that these were always equal to the standard derivative, or at least to a finite difference estimate of the derivative. Then it became clear that this was only true sometimes; in particular, for problems (like this one) in which the shape of the flow region changed, the sensitivities and the derivatives can differ by a quantity that measures the effects of geometry changes.

Once I realized this, I had to figure out whether this term could be accounted for, and also estimate the effects and sizes of the various kinds of approximation error that occurred.

• thesis_1995_vt.tex
• thesis_1995_vt.pdf

• title.tex, the Title.
• abstract.tex, the Abstract.
• thanks.tex, the Thanks.
• dedication.tex, the Dedication.
• forematter.tex, the Forematter.
• chap01.tex, Chapter 01: The Forebody Simulator Problem.
• chap02.tex, Chapter 02: A Parameterized Class of Flow Problems.
• chap03.tex, Chapter 03: Continuous Fluid Flow Equations.
• chap04.tex, Chapter 04: Discrete Fluid Flow Equations.
• chap05.tex, Chapter 05: State Sensitivities with Respect to a Parameter.
• chap06.tex, Chapter 06: Sensitivities for an Explicit Parameter.
• chap07.tex, Chapter 07: Sensitivities for an Implicit Parameter.
• chap08.tex, Chapter 08: Finite Difference Estimates of Sensitivities.
• chap09.tex, Chapter 09: Optimization and the Cost Functional.
• chap10.tex, Chapter 10: A Poorly Scaled Optimization.
• chap11.tex, Chapter 11: A Discretized Sensitivity Failure.
• chap12.tex, Chapter 12: A Local Minimizer.
• chap13.tex, Chapter 13: Optimization with a Penalty Functional.
• chap14.tex, Chapter 14: Flows at Higher Reynolds Number.
• chap15.tex, Chapter 15: Approximating Nonfeasible Targets.
• chap16.tex, Chapter 16: Efficient Solution of the Optimization Problem.
• chap17.tex, Chapter 17: The Computational Algorithm.
• chap18.tex, Chapter 18: Summary and Conclusions.
• refer.tex, References.
• vita.tex, Vita.

Figures:

• fig_01.png, An engine and forebody simulator, in a wind tunnel;
• fig_02.png, An abstract model of the forebody simulator problem;
• fig_03.png, An inflow specified with eight parameters;
• fig_04.png, A bump specified with two parameters;
• fig_05.png, The flow problem (lambda, alpha(1), alpha(2), alpha(3), Re) = (1.0, 1.0, 0.5, 1.5, 10.0);
• fig_06.png, The velocity solution for the five parameter problem;
• fig_07.png, Contours of the pressure solution for the five parameter problem;
• fig_08.png, Velocity vectors and pressure contour lines for Poiseuille flow;
• fig_09.png, Decomposition of a flow region into nodes;
• fig_10.png, A Taylor Hood element;
• fig_11.png, Decomposition of a flow region into elements;
• fig_12.png, Discrete velocity sensitivity with respect to the inflow parameter lambda;
• fig_13.png, Discrete velocity sensitivity with respect to the Re parameter;
• fig_14.png, Estimating the solution at a moving point;
• fig_15.png, How an integral changes when the integrand and region both vary;
• fig_16.png, Discretized velocity alpha-sensitivity;
• fig_17.png, Discretized alpha-sensitivity nodal values;
• fig_18.png, The corresponding alpha-finite difference nodal values;
• fig_19.png, Adjusted finite difference alpha-sensitivity nodal values;
• fig_20.png, Discretized sensitivity/adjusted finite coefficient difference discrepancies;
• fig_21.png, The flow region and velocity field for the target parameters;
• fig_22.png, The bump and velocity field for the target parameters;
• fig_23.png, The computed bump and velocity field using discretized sensitivities;
• fig_24.png, The computed bump and velocity field using a finite difference cost gradient;
• fig_25.png, The bump and velocity using finite differences for sensitivities;
• fig_26.png, Discretized velocity alpha(2)-sensitivity field;
• fig_27.png, The target flow region, velocity field, and profile line;
• fig_28.png, The flow produced with discretized sensitivities;
• fig_29.png, Values of Jh2(beta(S)) along a line;
• fig_30.png, Values of ( del J2) h (beta(S)) * dS along a line;
• fig_31.png, A contour map of the cost functional Jh2 in a plane;
• fig_32.png, Jh2 gradient directions by discretized sensitivities;
• fig_33.png, Jh2 gradient directions by finite difference sensitivities;
• fig_34.png, Comparison of Jh2 gradient directions;
• fig_35.png, The local optimizing flow;
• fig_36.png, Values of Jh2 on a line between the local and global minimizers;
• fig_37.png, Contours of Jh2 on a plane containing the two minimizers;
• fig_38.png, Contours and normalized gradients of Jh2;
• fig_39.png, The flow for h = 0.166;
• fig_40.png, The flow for h = 0.125;
• fig_41.png, Contours for cost functional Jh3 with epsilon = 0.0002;
• fig_42.png, Values of Jh2 during the optimization of Jh3;
• fig_43.png, Velocity sensitivities for Re = 1;
• fig_44.png, Velocity sensitivities for Re = 10;
• fig_45.png, Velocity sensitivities for Re = 100;
• fig_46.png, Velocity sensitivities for Re = 500;
• fig_47.png, Velocity sensitivities for Re = 1000;
• fig_48.png, Example 1: The target flow;
• fig_49.png, Example 1: The optimizing flow;
• fig_50.png, Example 1: Comparison of target and achieved flows;
• fig_51.png, Example 2: The target flow;
• fig_52.png, Example 2: The optimizing flow;
• fig_53.png, Example 2: Comparison of the target and minimizing flows;
• fig_54.png, Example 2: Comparison of the target and linear interpolant flows;
• fig_55.png, Example 3: The "artificial" flow;
• fig_56.png, Example 3: The optimizing flow, near the profile line;
• fig_57.png, Example 3: The optimizing inflow (includes outflow!);
• fig_58.png, Example 3: The match between target and desired flows;
• fig_59.png, CPU time as a function of Newton tolerance, Re = 1;