1. P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2014 – 35148: Continuous Solution for Boundary Value Problems on Non Continuous Geometry 34 th CIE, BUFFALO, NY, Aug 2014 P. Venkataraman

2. Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2014 – 35148: Continuous Solution for Boundary Value Problems on Non Continuous Geometry 34 th CIE, BUFFALO, NY, Aug 2014 TODAY’S PRESENTATION 1.MOTIVATION 2.BEZIER FUNCTIONAL REPRESENTATION 3.EXAMPLE 1: LAPLACE EQUATION 4.EXAMPLE 2: POISSON EQUATION 5.EXAMPLE 3: EXPERIMENTS WITH NONLINEAR PDE 6.CONCLUSIONs

3. P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2014 – 35148: Continuous Solution for Boundary Value Problems on Non Continuous Geometry 34 th CIE, BUFFALO, NY, Aug 2014 3 Motivation I Boundary Value Problems (BVP) on non-rectangular and non-continuous domain are mostly solved by Domain discretization techniques (finite element, finite volume, finite difference, etc.) Analytical solutions when they exist provide : Single solution over the domain Continuous higher order derivatives Analytical computation of incidental data based on the continuous solution Analytical Solutions of such problems are difficult and rare, except in the case of domain with special properties and if the BVP is linear.

4. P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2014 – 35148: Continuous Solution for Boundary Value Problems on Non Continuous Geometry 34 th CIE, BUFFALO, NY, Aug 2014 4 Motivation II This paper identifies the solution of BVP on a non-rectangular and non- continuous domains, using functional approximation. The functions used are the Bezier functions (based on Bernstein Polynomials) In essence, this is a meshless approach that provides a single continuous solution over the domain, a solution with continuous higher derivatives, and a solution that does not care if the problem is single, coupled, linear, or nonlinear BVP The method is: Direct Simple Requires no transformation of the problem (strong form of BVP) The solution (over the entire domain) is available in polynomial form (closed)

5. P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2014 – 35148: Continuous Solution for Boundary Value Problems on Non Continuous Geometry 34 th CIE, BUFFALO, NY, Aug 2014 5 Motivation III The solution of the BVP can be obtained either using a standard least squared error measure (LSE) or a least absolute error measure (LAE), in both the residuals and the boundary conditions Solution is determined at discrete points in the interior and the boundary. These points must be representative and are not required to be dense The solution depends on the order of the function chosen apriori and therefore can only be considered approximate. However changes in order changes the solution in a small way. Therefore, the solution can be considered robust. Continuous solutions of the linear BVP over a non-rectangular domain and non-continuous are usually not available, as far as the author can ascertain. The challenge of a continuous solution to a general nonlinear BVP over such a domain is rarely contemplated

6. P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2014 – 35148: Continuous Solution for Boundary Value Problems on Non Continuous Geometry 34 th CIE, BUFFALO, NY, Aug 2014 6 Bezier Function Representation 1 For this paper, the Bernstein basis representation of the Bezier function, using two parameters, (r, s) is Each B i,j represents a set three values, defining a vertex location in three- dimensional Euclidean space. m is the order of the surface ( also the polynomial) in x- direction. n is the order of the surface ( also the polynomial) in the y – direction. J m,i and K n, j are the Bernstein basis functions. The use of the Bézier function guarantees the existence of a bounded real valued function provided the vertices are bounded

7. P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2014 – 35148: Continuous Solution for Boundary Value Problems on Non Continuous Geometry 34 th CIE, BUFFALO, NY, Aug 2014 7 Bezier Function Representation 2 A mixture of symbolic and numeric computation is used for computation. The unique nature of the solution was considered more important than speed The formulation of the error is through symbolic calculation The minimization of the error is accomplished numerically In this paper we linearly relate the parameters r and s to the independent variables x and y. This transformation allows us to generate the derivatives of the functions required for the BVP and the boundary conditions directly without employing chain rule The translation from symbolic to numeric objective function for the optimizer is done using a special built-in matlabFunction function. The solution was obtained through the unconstrained optimizer fminunc and fmincon from the MATLAB Optimization Toolbox

8. P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2014 – 35148: Continuous Solution for Boundary Value Problems on Non Continuous Geometry 34 th CIE, BUFFALO, NY, Aug 2014 8 Example 1: POISSON’S EQUATION 1 The first problem is the solution to Laplace equation over the domain show below with the boundary conditions (with k = 1, Q = 0, q x and q y are 3) Bezier Points Residual Points Boundary Points.

9. P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2014 – 35148: Continuous Solution for Boundary Value Problems on Non Continuous Geometry 34 th CIE, BUFFALO, NY, Aug 2014 9 Example 1: LAPLACE EQUATION 2 Objective function : Number of Bézier points is 121 (for function of order 10 in both directions). Number of points on the boundary (n B ) was 68. Number of total points for the error in the residuals (n R ) were 468.

10. P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2014 – 35148: Continuous Solution for Boundary Value Problems on Non Continuous Geometry 34 th CIE, BUFFALO, NY, Aug 2014 10 Example 1: LAPLACE EQUATION 3 Bezier Solution Random Initial Guess COMSOL Solution Bezier Solution Structured Initial Guess

11. P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2014 – 35148: Continuous Solution for Boundary Value Problems on Non Continuous Geometry 34 th CIE, BUFFALO, NY, Aug 2014 11 Example 1: LAPLACE EQUATION 4 Boundary Error Residual Error Average Absolute Residual0.051 Average Boundary Error2.3

12. P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2014 – 35148: Continuous Solution for Boundary Value Problems on Non Continuous Geometry 34 th CIE, BUFFALO, NY, Aug 2014 12 Example 2: POISSON EQUATION 1 The second example is the solution to Poisson equation over the same domain with the same boundary conditions (with k = 1, Q = 25, q x and q y are 3) Objective function :

13. P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2014 – 35148: Continuous Solution for Boundary Value Problems on Non Continuous Geometry 34 th CIE, BUFFALO, NY, Aug 2014 13 Example 2: POISSON EQUATION 2 COMSOL Solution Bezier Solution Structured Initial Guess Number of Bézier points is 121 (for function of order 10 in both directions). Number of points on the boundary (n B ) was 68. Number of total points for the error in the residuals (n R ) were 468.

14. P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2014 – 35148: Continuous Solution for Boundary Value Problems on Non Continuous Geometry 34 th CIE, BUFFALO, NY, Aug 2014 14 Example 1: POISSON EQUATION 3 Boundary Error Residual Error Average Residual Error0.027 Average Boundary Error2.69

15. P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2014 – 35148: Continuous Solution for Boundary Value Problems on Non Continuous Geometry 34 th CIE, BUFFALO, NY, Aug 2014 15 Example 3: EXPLORING NON LINEAR PDE1 The final example is the solution to non-linear PDE over the same domain with the same boundary conditions (with k = 1, Q = 25, q x and q y are 3) Solution was not satisfactory with unconstrained optimization as the boundary error and the residuals had opposite effects

16. P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2014 – 35148: Continuous Solution for Boundary Value Problems on Non Continuous Geometry 34 th CIE, BUFFALO, NY, Aug 2014 16 Example 3: A Nonlinear Equation 2 Using Constrained Optimization to relax constraints Average residual error1128 Average boundary error0.014

17. P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2014 – 35148: Continuous Solution for Boundary Value Problems on Non Continuous Geometry 34 th CIE, BUFFALO, NY, Aug 2014 17 Example 3: A Nonlinear Equation 3 Average residual error7.97 Average boundary error2.94

18. P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2014 – 35148: Continuous Solution for Boundary Value Problems on Non Continuous Geometry 34 th CIE, BUFFALO, NY, Aug 2014 18 1. The formulation is simple Conclusions 2. The set up is direct 3. Meshless (no domain discretization) 4. Partial differential equations handled in original form 5. Procedure is independent of type or class of problems 6. A single continuous solution over the entire domain 7. Number of points for error computation is not important 8. A mix of symbolic and numeric computation for error control 9. The procedure finds approximate analytical solutions for difficult BVP

19. P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2014 – 35148: Continuous Solution for Boundary Value Problems on Non Continuous Geometry 34 th CIE, BUFFALO, NY, Aug 2014 19 Future Work 1. A return to unconstrained optimization for non-linear PDE using selective location of Bezier coefficients 2. Incorporating analytical gradients since they are available 3. Develop a new optimization technique to overcome slow convergence that are natural for Bezier functions 4. Extend the approach to fluid mechanics that involve coupled non-linear PDE

20. P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2014 – 35148: Continuous Solution for Boundary Value Problems on Non Continuous Geometry 34 th CIE, BUFFALO, NY, Aug 2014 20 Questions ?