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The use of adjoint methods is common in computational fluid dynamics to reduce the cost of the sensitivity analysis in an optimization cycle. The forward mode ADjoint is a combination of an adjoint sensitivity analysis method with a forward mode automatic differentiation (AD) and is a modification of the reverse mode ADjoint method proposed by \citet{Mader:2008:B}. A colouring acceleration technique is presented to reduce the computational cost increase associated with forward mode AD. The forward mode AD facilitates the implementation of the laminar Navier--Stokes (NS) equations. The forward mode ADjoint method is applied to a three-dimensional computational fluid dynamics solver. The resulting Euler and viscous ADjoint sensitivities are compared to the reverse mode Euler ADjoint derivatives and a complex-step method to demonstrate the reduced computational cost and accuracy. Both comparisons demonstrate the benefits of the colouring method and the practicality of using a forward mode AD.

The use of adjoint methods is common in computational fluid dynamics to reduce the cost of the sensitivity analysis in an optimization cycle. The forward mode ADjoint is a combination of an adjoint sensitivity analysis method with a forward mode automatic differentiation (AD) and is a modification of the reverse mode ADjoint method proposed by \citet{Mader:2008:B}. A colouring acceleration technique is presented to reduce the computational cost increase associated with forward mode AD. The forward mode AD facilitates the implementation of the laminar Navier--Stokes (NS) equations. The forward mode ADjoint method is applied to a three-dimensional computational fluid dynamics solver. The resulting Euler and viscous ADjoint sensitivities are compared to the reverse mode Euler ADjoint derivatives and a complex-step method to demonstrate the reduced computational cost and accuracy. Both comparisons demonstrate the benefits of the colouring method and the practicality of using a forward mode AD.

We present a framework for modeling multibody systems based on the method of natural coordinates and Lagrange's equation of the first kind, resulting in a system of Differential-Algebraic Equations (DAEs). The C++ package DAETS (DAEs by Taylor Series), a robust high-index DAE solver, is utilized to solve the models. The simulation process is straightforward, with no need to derive equations of motion directly. Instead, the user supplies a Lagrangian, kinematic constraints, and if applicable, a dissipation function and external forces. A corresponding system of DAEs is formed by computing the required derivatives via automatic differentiation. DAETS primarily uses Cartesian coordinates as variables, eliminating angles and the associated trigonometric functions, which results in simplified models. Furthermore, DAETS provides direct access to the position/velocity data of any desired points or vectors as output, facilitating post-processing tasks, such as visualization. The main focus of this thesis is on establishing the viability of our framework through case studies. We simulate seven multibody systems and compare our results with those of reference models developed in the Simulink environment of MATLAB. A detailed account of the modeling process is given for each system, demonstrating the ease and intuitiveness of our approach. We also provide, from both DAETS and Simulink, the time history plots of several position coordinates to allow for direct comparison. Finally, we compute two types of errors over time. Our findings show that the results of DAETS match those of the reference models under different error tolerances for the studied systems, indicating that our framework is capable of simulating a wide variety of mechanisms with a superb degree of accuracy. / Thesis / Master of Science (MSc)

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This work focuses on the development of efficient methods for wing shape optimization for morphing wing technologies. Existing wing shape optimization processes typically rely on computational fluid dynamics tools for aerodynamic analysis, but the computational cost of these tools makes optimization of all but the most basic problems intractable. In this work, we present a set of tools that can be used to efficiently explore the design spaces of morphing wings without reducing the fidelity of the results significantly. Specifically, this work discusses automatic differentiation of an aerodynamic analysis tool based on lifting line theory, a light-weight gradient-based optimization framework that provides a parallel function evaluation capability not found in similar frameworks, and a modification to the lifting line equations that makes the analysis method and optimization process suitable to wings of arbitrary aspect ratio. The toolset discussed is applied to several wing shape optimization problems. Additionally, a method for visualizing the design space of a morphing wing using this toolset is presented. As a result of this work, a light-weight wing shape optimization method is available for analysis of morphing wing designs that reduces the computational cost by several orders of magnitude over traditional methods without significantly reducing the accuracy of the results.

We propose two efficient automatic differentiation (AD) schemes to compute incrementally Taylor series and System Jacobian for solving differential-algebraic equations (DAEs) by Taylor series. Our schemes are based on topological ordering of a DAE's computational graph and then partitioning the topologically sorted nodes using structural information obtained from the DAE. Solving a DAE by Taylor series is carried out in stages. From one stage to another, partitions of the computational graph are incrementally activated so that we can reuse Taylor coefficients and gradients computed in previous stages. As a result, the computational complexity of evaluating a System Jacobian is independent of the number of stages. We also develop a common subexpression elimination (CSE) method to build a compact computational graph through operator overloading. The CSE method is of linear time complexity, which makes it suitable as a preprocessing step for general operator overloaded computing. By applying CSE, all successive overloaded computation can save time and memory. Furthermore, the computational graph of a DAE reveals its internal sparsity structure. Based on it, we devise an algorithm to propagate gradients in the forward mode of AD using compressed vectors. This algorithm can save both time and memory when computing the System Jacobian for sparse DAEs. We have integrated our approaches into the \daets solver. Computational results show multiple-fold speedups against two popular AD tools, \FAD~and ADOL-C, when solving various sparse and dense DAEs. / Thesis / Master of Science (MSc)

Adaptivity in both space and time has become the norm for solving problems modeled by partial differential equations. The size of the discretized problem makes uniformly refined grids computationally prohibitive. Adaptive refinement of meshes and time steps allows to capture the phenomena of interest while keeping the cost of a simulation tractable on the current hardware. Many fields in science and engineering require the solution of inverse problems where parameters for a given model are estimated based on available measurement information. In contrast to forward (regular) simulations, inverse problems have not extensively benefited from the adaptive solver technology. Previous research in inverse problems has focused mainly on the continuous approach to calculate sensitivities, and has typically employed fixed time and space meshes in the solution process. Inverse problem solvers that make exclusive use of uniform or static meshes avoid complications such as the differentiation of mesh motion equations, or inconsistencies in the sensitivity equations between subdomains with different refinement levels. However, this comes at the cost of low computational efficiency. More efficient computations are possible through judicious use of adaptive mesh refinement, adaptive time steps, and the discrete adjoint method. This dissertation develops a complete framework for fully discrete adjoint sensitivity analysis and inverse problem solutions, in the context of time dependent, adaptive mesh, and adaptive step models. The discrete framework addresses all the necessary ingredients of a state–of–the–art adaptive inverse solution algorithm: adaptive mesh and time step refinement, solution grid transfer operators, a priori and a posteriori error analysis and estimation, and discrete adjoints for sensitivity analysis of flux–limited numerical algorithms. / Ph. D.

Derivative pricing, model calibration, and sensitivity analysis are the three main problems in financial modeling. The purpose of this study is to present an algorithm to improve the pricing process, the calibration process, and the sensitivity analysis of the double Heston model, in the sense of accuracy and efficiency. Using the optimized caching technique, our study reduces the pricing computation time by about 15%. Another contribution of this thesis is: a novel application of the Automatic Differentiation (AD) algorithms in order to achieve a more stable, more accurate, and faster sensitivity analysis for the double Heston model (compared to the classical finite difference methods). This thesis also presents a novel hybrid model by combing the heuristic method Differentiation Evolution, and the gradient method Levenberg--Marquardt algorithm. Our new hybrid model significantly accelerates the calibration process.

A computational tool that uses an automated and reliable procedure for systematic generation of global phase equilibrium diagram (GPED) is developed for binary system using equation of state and its extension to the ternary system is discussed. The proposed algorithm can handle solid phase and also can predict all major six types of phase diagrams. The procedure enables automatic generation of GPED which incorporates calculations of all important landmarks such as critical endpoints, quadruple point (if any), critical line, liquid-liquid-vapor line (if any), solid-liquid-liquid line (if any) and solid-liquid-vapor line. The method is also capable of locating all azeotropic phenomena such as azeotropic endpoint, critical azeotrope, pure azeotropic point and azeotropic lines. Although, we demonstrated the methodology for cubic equation of state, the proposed strategy is completely general that doesn't require any knowledge about the type of phase diagram and can be applied to any pressure explicit equation of state model. Newton homotopy based global method has been applied for phase stability test and critical point calculations to ensure reliability. Having computed the binary phase diagrams, the methodology to generate global phase diagrams for ternary system is discussed that can locate all important thermodynamic landmarks such as tricritical point, quadruple critical endpoint, quadruple azeotropic endpoint, quintuple point and critical azeotropic endpoint. The procedure to trace ternary phenomena having two degree of freedom such as critical surface, solid-liquid-vapor surface and liquid-liquid-vapor surface has been discussed. Finally, applications of reliable global methods to solve the fluid-fluid phase equilibrium problem using SAFT equation for binary system and the solid-fluid phase equilibrium problem for binary and ternary systems have been demonstrated through representative computations.