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P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2013 – 12269: Continuous Solution for Boundary Value Problems on Nonrectangular Geometry 33 nd CIE, PORTLAND, OR, Aug 2013 P. Venkataraman

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Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2013 – 12269: Continuous Solution for Boundary Value Problems on Nonrectangular Geometry 33 nd CIE, PORTLAND, OR, Aug 2013 TODAY’S PRESENTATION 1.MOTIVATION 2.BEZIER FUNCTIONAL REPRESENTATION 3.EXAMPLE 1: POISSON’S EQUATION 4.EXAMPLE 2: LAPLACE EQUATION 5.EXAMPLE 3: NONLINEAR PDE 6.CONCLUSION

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P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2013 – 12269: Continuous Solution for Boundary Value Problems on Nonrectangular Geometry 33 nd CIE, PORTLAND, OR, Aug 2013 3 Motivation I Boundary Value Problems (BVP) on rectangular (regular) domain can be solved either by (a)Domain discretization techniques (finite element, finite volume, finite difference ), or, (b)Non-discretization techniques (meshless, analytical, using function approximation – adopted in this paper) The advantage of the particular functional representation of this paper allows extraction of additional properties of the data that may not be obvious Single solution over the domain Continuous higher order derivatives Analytical computation of incidental data based on the continiuos solution Does not care if the system is linear, nonlinear, ordinary, partial, single, or coupled systems of differential equations

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P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2013 – 12269: Continuous Solution for Boundary Value Problems on Nonrectangular Geometry 33 nd CIE, PORTLAND, OR, Aug 2013 4 Motivation II This paper illustrates the solution of BVP on a nonrectangular (irregular) domains, using functional approximation through the Bezier functions (or Bernstein Polynomials) Currently such problems are only solved using domain discretization techniques In essence, this is a meshless approach that provides all of the advantages mentioned in the previous slide and in addition 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)

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P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2013 – 12269: Continuous Solution for Boundary Value Problems on Nonrectangular Geometry 33 nd CIE, PORTLAND, OR, Aug 2013 5 Motivation III The solution of the BVP is obtained using a standard least square error measure(or absolute error measure) in both the residuals and the boundary conditions Solution is determined at discrete points in the interior and the boundary The solution depends on the order of the function and therefore can only be considered approximate. However variation in the parameters of the method only changes the solution in a small way. Therefore, the solution can be considered robust Continuous solutions of the linear BVP over a nonrectangular domain are usually not available, as far as the author can ascertain. The challenge of a continuous solution to a general nonlinear BVP over the same region is also a state of the art

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P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2013 – 12269: Continuous Solution for Boundary Value Problems on Nonrectangular Geometry 33 nd CIE, PORTLAND, OR, Aug 2013 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 or polynomial form. The use of the Bézier function guarantees the existence of a bounded real valued function (provided the vertices are bounded) The numerical optimizer will determine design variables that are bounded. Therefor an approximate solution to the BVP problem will always exist even if its quality is wanting.

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P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2013 – 12269: Continuous Solution for Boundary Value Problems on Nonrectangular Geometry 33 nd CIE, PORTLAND, OR, Aug 2013 7 Bezier Function Representation 2 A mixture of symbolic and numeric computation is used for computation, The formulation of the error is through symbolic calculation The minimization of the error is accomplished numerically We linearly relate the parameters r and s to the independent variables x and y We take advantage of this transformation to generate the higher derivatives of the functions used in the BVP, and if necessary, for Neumann boundary conditions. The translation from symbolic to numeric objective function for the optimizer is done using a special built-in matlabFunction function. The solution was discovered through the unconstrained optimizer fminunc from the MATLAB Optimization Toolbox

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P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2013 – 12269: Continuous Solution for Boundary Value Problems on Nonrectangular Geometry 33 nd CIE, PORTLAND, OR, Aug 2013 8 Example 1: POISSON’S EQUATION 1 The first problem is the solution to Poisson’s equation over a circular domain Points used to generate the solution Points used to calculate residuals Points used to calculate boundary error

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P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2013 – 12269: Continuous Solution for Boundary Value Problems on Nonrectangular Geometry 33 nd CIE, PORTLAND, OR, Aug 2013 9 Example 1: POISSON’S EQUATION 2 Objective function : Number of Bézier points is 36 (for function of order 5). Number of points on the boundary (n B ) was 21. Number of total points for the error in the residuals (n R ) were 278. m (x-order)n (y-order)fiterations 444.12e-01198 332.02e-01237 223.9e-01326 The problem has an analytical solution of second order

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P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2013 – 12269: Continuous Solution for Boundary Value Problems on Nonrectangular Geometry 33 nd CIE, PORTLAND, OR, Aug 2013 10 Example 1: POISSON’S EQUATION 3 Bezier Solution (5, 5) COMSOL Solution

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P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2013 – 12269: Continuous Solution for Boundary Value Problems on Nonrectangular Geometry 33 nd CIE, PORTLAND, OR, Aug 2013 11 Example 2: LAPLACE EQUATION 1 This example deals with Laplace equation over a five sided region There are 4 straight edges and 1 quarter circle The boundary conditions at the edges are also detailed on the figure For discontinuous solution (FEM), the temperatures at the intersection of the edges can have different temperatures on the two edges. That is different temperatures at the same point For continuous solution on the domain, the same point cannot have two different values – therefore solutions will violate discontinuous boundary conditions. The figure above indicates the modified boundary conditions

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P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2013 – 12269: Continuous Solution for Boundary Value Problems on Nonrectangular Geometry 33 nd CIE, PORTLAND, OR, Aug 2013 12 Example 2: LAPLACE EQUATION 2 Bezier points (100) Residual points (260) Boundary points (50) m = 9, n = 9 TotalAverage Residual13.40.05 Boundary83817 Solution

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P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2013 – 12269: Continuous Solution for Boundary Value Problems on Nonrectangular Geometry 33 nd CIE, PORTLAND, OR, Aug 2013 13 Example 2: LAPLACE EQUATION 3 Continuous Solution COMSOL Solution Boundary Error Residual Error

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P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2013 – 12269: Continuous Solution for Boundary Value Problems on Nonrectangular Geometry 33 nd CIE, PORTLAND, OR, Aug 2013 14 Example 3: A Nonlinear Equation 1 A nonlinear example on the same domain with same boundary conditions The only change required is to incorporate the new residual function Everything else remained the same – including the starting guesses and the optimizer. Generally: errors are bigger more iterations solution moves to a local minimum Objective function can be the least squared error in the residuals and the boundary conditions – OR – least sum of absolute error in the residuals and boundary conditions

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P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2013 – 12269: Continuous Solution for Boundary Value Problems on Nonrectangular Geometry 33 nd CIE, PORTLAND, OR, Aug 2013 15 Example 3: A Nonlinear Equation 2 No solution is available for comparison. Therefore two solutions are shown Structured Initial Guess Final Solution Sum of Absolute Error Initial7.6e07 Final31128 Average Error Residual1.1 Boundary18

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P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2013 – 12269: Continuous Solution for Boundary Value Problems on Nonrectangular Geometry 33 nd CIE, PORTLAND, OR, Aug 2013 16 Example 3: A Nonlinear Equation 3 Random Initial Guess Final Solution Sum of Absolute Error Initial8.6e08 Final30471 Average Error Residual1.5 Boundary19.1 Final Solution (structured) Therefore we have a mesh free procedure to obtain a continuous solution to PDEs, on a non-rectangular domain, irrespective of the linear or nonlinear nature of the PDE.

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P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2013 – 12269: Continuous Solution for Boundary Value Problems on Nonrectangular Geometry 33 nd CIE, PORTLAND, OR, Aug 2013 17 1. The formulation is simple Conclusions 2. The set up is direct 3. Meshless (no domain discretization) 4. Differential equations handled in original form 5. Exact derivatives in residual computation 6. Standard unconstrained optimizer 7. Procedure is independent of type or class of problems 8. A single continuous solution over the entire domain 9. Number of points for error computation is not important 10. A mix of symbolic and numeric computation for error control 11. The procedure provides decent approximate solutions for difficult BVP

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P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2013 – 12269: Continuous Solution for Boundary Value Problems on Nonrectangular Geometry 33 nd CIE, PORTLAND, OR, Aug 2013 18 1. Incorporate analytical gradients because the formulation is symbolic to improve and speed convergence Future Work 2. To investigate separable solutions to take advantage of the excellent blending properties of the Bernstein basis 3. Reduce dimension of the problem through separable solutions 4. Extend these investigations to Inverse problems in non rectangular domains

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P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2013 – 12269: Continuous Solution for Boundary Value Problems on Nonrectangular Geometry 33 nd CIE, PORTLAND, OR, Aug 2013 19 Questions ?

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