Retrieving sea ice drag coefficients and turning angles from in situ and satellite observations using an inverse modeling framework

Author Posting. © American Geophysical Union, 2019. This article is posted here by permission of American Geophysical Union for personal use, not for redistribution. The definitive version was published in Journal of Geophysical Research-Oceans 124(8), (2019): 6388-6413, doi: 10.1029/2018JC014881.For ice concentrations less than 85%, internal ice stresses in the sea ice pack are small and sea ice is said to be in free drift. The sea ice drift is then the result of a balance between Coriolis acceleration and stresses from the ocean and atmosphere. We investigate sea ice drift using data from individual drifting buoys as well as Arctic‐wide gridded fields of wind, sea ice, and ocean velocity. We perform probabilistic inverse modeling of the momentum balance of free‐drifting sea ice, implemented to retrieve the Nansen number, scaled Rossby number, and stress turning angles. Since this problem involves a nonlinear, underconstrained system, we used a Monte Carlo guided search scheme—the Neighborhood Algorithm—to seek optimal parameter values for multiple observation points. We retrieve optimal drag coefficients of CA=1.2×10−3 and CO=2.4×10−3 from 10‐day averaged Arctic‐wide data from July 2014 that agree with the AIDJEX standard, with clear temporal and spatial variations. Inverting daily averaged buoy data give parameters that, while more accurately resolved, suggest that the forward model oversimplifies the physical system at these spatial and temporal scales. Our results show the importance of the correct representation of geostrophic currents. Both atmospheric and oceanic drag coefficients are found to decrease with shorter temporal averaging period, informing the selection of drag coefficient for short timescale climate models.The scripts developed for this publication are available at the GitHub (https://github.com/hheorton/Freedrift_inverse_submit). The Neighborhood Algorithm was developed and kindly supplied by M. Sambridge (http://www.iearth.org.au/codes/NA/). Ice‐Tethered Profiler data are available via the Ice‐Tethered Profiler program website (http://whoi.edu/itp). Buoy data were collected as part of the Marginal Ice Zone program (www.apl.washington.edu/miz) funded by the U.S. Office of Naval Research. The ice drift data were kindly supplied by N. Kimura. H. H. was funded by the Natural Environment Research Council (Grants NE/I029439/1 and NE/R000263/1). M. T. was partially funded by the SKIM Mission Science Study (SKIM‐SciSoc) Project ESA RFP 3‐15456/18/NL/CT/gp. T. A. was supported at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. M. T. and H. H. thank Dr. Nicolas Brantut for early discussions on the implementation of inverse modeling techniques.2020-02-1

Differential-Algebraic Equations

Differential-Algebraic Equations (DAE) are today an independent field of research, which is gaining in importance and becoming of increasing interest for applications and mathematics itself. This workshop has drawn the balance after about 25 years investigations of DAEs and the research aims of the future were intensively discussed

Optimization with block variables: theory and applications.

本博士论文对于有结构但又有相当一般性的约束条件下的非线性优化问题给出了系统性研究。比较经典的例子包指球面约束下的多重线性函数优化问题。这些模型已被广泛应用于数值线性代数、材料科学、量子物理学、信号处理、语音识别、生物医学工程以及控制论等。本论文着重探讨一类特定的方法来解这些广义模型，即块变量改进方法。具体地说，我们构造了一类块坐标下降型搜索算法来解带块变量结构的非线性优化问题。这类算法通过每次迭代中只更新一块变量以达到最大限度的目标函数值的改进(因而，这一新搜索算法命名为最优块改进算法(简称为MBI)。之后，我们重点研究了该算法在求解众多领域中实际问题的潜在能力。首先，这一算法可以直接应用于生物信息学中聚类基因表达数据的一种新模型的设计及求解。接着，我们把注意力转移到球面约束下的齐次多项式优化问题，此问题与张量的最优秩-1 逼近问题相关。对于这一优化问题， MBI 算法通常可以在较少的计算时间内找到全局最优解。第三，我们继续深入研究多项式优化问题，在双协半正定的新概念下建立了齐次多项式优化问题与其多重线性优化问题关系的一般性结果。最后，我们在Tucker 分解的框架下给出了求解高阶张量的最优多重线性秩的逼近问题的方法，并提出一种新的模型和算法来解未知变量数的Tucker 分解问题。本论文讨论并试验了一些应用实例，数值实验表明所提出的算法分别对于求解以上这些问题是可行并有效的。In this thesis we present a systematic analysis for optimization of a general nonlinear function, subject to some fairly general constraints. A typical example includes the optimization of a multilinear tensor function over spherical constraints. Such models have found wide applications in numerical linear algebra, material sciences, quantum physics, signal processing, speech recognition, biomedical engineering, and control theory. This thesis is mainly concerned with a specific approach to solve such generic models: the block variable improvement method. Specifically, we establish a block coordinate descent type search method for nonlinear optimization, which accepts only a block update that achieves the maximum improvement (hence the name of our new search method: maximum block improvement (MBI)). Then, we focus on the potential capability of this method for solving problems in various research area. First, we demonstrate that this method can be directly used in designing a new framework for co-clustering gene expression data in the area of bioinformatics. Second, we turn our attention to the spherically constrained homogeneous polynomial optimization problem, which is related to best rank-one approximation of tensors. The MBI method usually finds the global optimal solution at a low computational cost. Third, we continue to consider polynomial optimization problems. A general result between homogeneous polynomials and multi-linear forms under the concept of co-quadratic positive semidefinite is established. Finally, we consider the problem of finding the best multi-linear rank approximation of a higher-order tensor under the framework of Tucker decomposition, and also propose a new model and algorithms for computing Tucker decomposition with unknown number of components. Some real application examples are discussed and tested, and numerical experiments are reported to reveal the good practical performance and efficiency of the proposed algorithms for solving those problems.Detailed summary in vernacular field only.Chen, Bilian.Thesis (Ph.D.)--Chinese University of Hong Kong, 2012.Includes bibliographical references (leaves 86-98).Abstract also in Chinese.Abstract --- p.iiiAcknowledgements --- p.viiChapter 1 --- Introduction --- p.1Chapter 1.1 --- Overview --- p.1Chapter 1.2 --- Notations and Preliminaries --- p.5Chapter 1.2.1 --- The Tensor Operations --- p.6Chapter 1.2.2 --- The Tensor Ranks --- p.9Chapter 1.2.3 --- Polynomial Functions --- p.11Chapter 2 --- The Maximum Block Improvement Method --- p.12Chapter 2.1 --- Introduction --- p.12Chapter 2.2 --- MBI Method and Convergence Analysis --- p.14Chapter 3 --- Co-Clustering of Gene Expression Data --- p.19Chapter 3.1 --- Introduction --- p.19Chapter 3.2 --- A New Generic Framework for Co-Clustering Gene Expression Data --- p.22Chapter 3.2.1 --- Tensor Optimization Model of The Co-Clustering Problem --- p.22Chapter 3.2.2 --- The MBI Method for Co-Clustering Problem --- p.23Chapter 3.3 --- Algorithm for Co-Clustering 2D Matrix Data --- p.25Chapter 3.4 --- Numerical Experiments --- p.27Chapter 3.4.1 --- Implementation Details and Some Discussions --- p.27Chapter 3.4.2 --- Testing Results using Microarray Datasets --- p.30Chapter 3.4.3 --- Testing Results using 3D Synthesis Dataset --- p.32Chapter 4 --- Polynomial Optimization with Spherical Constraint --- p.34Chapter 4.1 --- Introduction --- p.34Chapter 4.2 --- Generalized Equivalence Result --- p.37Chapter 4.3 --- Spherically Constrained Homogeneous Polynomial Optimization --- p.41Chapter 4.3.1 --- Implementing MBI on Multilinear Tensor Form --- p.42Chapter 4.3.2 --- Relationship between Homogeneous Polynomial Optimization over Spherical Constraint and Tensor Relaxation Form --- p.43Chapter 4.3.3 --- Finding a KKT point for Homogeneous Polynomial Optimization over Spherical Constraint --- p.45Chapter 4.4 --- Numerical Experiments on Randomly Simulated Data --- p.47Chapter 4.4.1 --- Multilinear Tensor Function over Spherical Constraints --- p.49Chapter 4.4.2 --- Tests of Another Implementation of MBI --- p.49Chapter 4.4.3 --- General Polynomial Function over Quadratic Constraints --- p.51Chapter 4.5 --- Applications --- p.53Chapter 4.5.1 --- Rank-One Approximation of Super-Symmetric Tensors --- p.54Chapter 4.5.2 --- Magnetic Resonance Imaging --- p.55Chapter 5 --- Logarithmically Quasiconvex Optimization --- p.58Chapter 5.1 --- Introduction --- p.58Chapter 5.2 --- Logarithmically Quasiconvex Optimization --- p.60Chapter 5.2.1 --- A Simple Motivating Example --- p.61Chapter 5.2.2 --- Co-Quadratic Positive Semide nite Tensor Form --- p.61Chapter 5.2.3 --- Equivalence at Maxima --- p.64Chapter 6 --- The Tucker Decomposition and Generalization --- p.68Chapter 6.1 --- Introduction --- p.68Chapter 6.2 --- Convergence of Traditional Tucker Decomposition --- p.71Chapter 6.3 --- Tucker Decomposition with Unknown Number of Components --- p.73Chapter 6.3.1 --- Problem Formulation --- p.74Chapter 6.3.2 --- Implementing the MBI Method on Tucker Decomposition with Unknown Number of Components --- p.75Chapter 6.3.3 --- A Heuristic Approach --- p.79Chapter 6.4 --- Numerical Experiments --- p.80Chapter 7 --- Conclusion and Recent Developments --- p.83Bibliography --- p.8

Nonlinear dynamics and numerical uncertainties in CFD

The application of nonlinear dynamics to improve the understanding of numerical uncertainties in computational fluid dynamics (CFD) is reviewed. Elementary examples in the use of dynamics to explain the nonlinear phenomena and spurious behavior that occur in numerics are given. The role of dynamics in the understanding of long time behavior of numerical integrations and the nonlinear stability, convergence, and reliability of using time-marching, approaches for obtaining steady-state numerical solutions in CFD is explained. The study is complemented with spurious behavior observed in CFD computations

항복점 현상으로 인한 금속 박판 재료의 꺾임 결함과 소부 경화에 대한 실험 및 수치해석 연구

학위논문(박사)--서울대학교 대학원 :공과대학 기계항공공학부,2020. 2. 김도년.The yield-point phenomenon of annealed or aged metals has duality in terms of avoiding it or utilizing it. Defects such as fluting in v-bending originated from the phenomenon can be avoided through roller-leveling process composed of multiple up-and-down bending operations. However, excessive leveling conditions can lead to superficies defects, and an adequate process condition remains still elusive. Utilizing the phenomenon, bake hardening characterized by the significant increase of yield stress after baking of pre-strained low carbon steel can be used for improving dent resistance in automotive sheet metal forming applications. However, many previous investigations about bake hardenability concentrate only on the bake hardening response of uniaxial tension with related influence factors, and numerous numerical studies for the dent resistance rarely consider bake hardening effect. To accurately predict the behavior of materials with this phenomenon, the constitutive model for computational elastoviscoplastic analysis should be able to depict the yield-point phenomenon, the Bauschinger effect, and bake hardenability, rendering it difficult to obtain a converged solution in implicit numerical analysis when the conventional one-point Newton method is used. In the present study, firstly, comprehensive experimental investigations are performed for the fluting defect in the v-bending process, its reduction by the roller-leveling process, and the dent resistance of an automotive bake hardenable steel. Systematic evaluation for the effect of roller-leveling condition on the fluting in v-bending is then carried out using pre-coated low carbon steel after examining the rate dependency and the cyclic characteristics of the phenomenon in uniaxial loads. The bake hardening behavior of a dual phase steel is observed in uniaxial load cases and static dent experiments conducted in pre-strained and bake hardened conditions. For numerical analysis to describe these experimental observation results, an implicit stress-integration procedure is formulated and implemented for a constitutive material model that can describe both the yield-point phenomenon and the Bauschinger effect. And we propose robust stress integration algorithms that can be used effectively in implicit finite element analysis employing the bisection method and the two-point Newton method. This material model is also integrated with a bake hardening model to illustrate bake hardening potentials. The validation results of the model with simple problems demonstrate that the model can be reliably used to calculate the solutions of the yield-point phenomenon problems that cannot be obtained using conventional iterative methods although these algorithms may require longer computational times. Numerical simulations corresponding to the experiments are carried out with material parameters determined to reproduce the uniaxial experiments. V-bending simulations at various roller-leveling conditions fairly demonstrate the fluting defect and its reduction experimentally observed. The bake hardening behaviors identified in the experiments are investigated in static dent simulations including a bake hardening step, and the bake hardening effect is overall described in numerical simulations. To conclude, the proposed analysis procedure is expected to be useful in estimating a proper leveling condition to prevent potential defects and dent resistance of automotive bake hardenable steels as well as investigating the effect of the yield-point phenomenon in various metal forming processes.소둔이 되었거나 시효가 발생한 금속의 항복점 현상은 이 현상을 회피하거나 활용하는 관점에서 양면성을 가지고 있다. 이 현상으로 발생하는 V굽힘 공정에서의 꺾임 결함은 소재에 상하방향 굽힘을 부과하는 롤러 레벨링 공정의 적용으로 감소될 수 있다. 롤러 레벨링 조건이 과할 경우 표면 결함이 발생할 수 있고, 적절한 공정 조건을 찾는 것이 여전히 어려운 실정이다. 예변형된 저탄소강을 구웠을 때 항복점이 현저하게 높이지는 특징을 보이는 소부 경화 거동은 이 현상을 활용하는 경우이며, 자동차 박판 금속 성형 응용 분야에서 덴트 저항성을 향상시키기 위해 사용된다. 그러나 소부 경화능에 대한 많은 연구는 일축 인장에서의 소부 경화 응답 및 그 영향인자에만 집중하고 있고, 자동차 강판의 덴트 저항성을 위한 수치적 연구는 소부 경화 효과를 거의 고려하지 않고 있다. 항복점 현상을 보이는 소재의 거동을 정확하게 예측하기 위해서는 점탄소성 수치해석을 위한 구성 모델이 항복점 현상, 바우싱거 효과, 그리고 소부 경화능을 묘사할 수 있어야 하지만, 이러한 예측과정에서 전통적인 1점 뉴턴법을 사용할 경우 내연적 수치해석에서 수렴된 해를 획득하기 어렵다. 본 연구에서는 먼저 V굽힘 공정에서의 꺾임, 롤러 레벨링 공정을 통한 꺾임 감소, 그리고 자동차 소부 경화 강판의 덴트 저항성에 대한 포괄적인 실험적 연구를 수행하였다. 일축 하중 하에서 저탄소 도장 강판의 속도 의존성과 주기 거동 특성을 조사한 뒤 V굽힘에서의 꺾임에 대한 롤러 레벨링 조건의 효과를 체계적으로 평가 하였다. 2상 강판의 소부 경화 거동은 예변형과 소부 경화 조건의 일축 하중 및 정적 덴트 실험에서 관찰되었다. 이러한 실험적 관찰 결과를 수치해석으로 묘사하기 위해, 항복점 현상과 바우싱거 효과를 동시에 묘사할 수 있는 재료 구성 모델에 대해 내연적 응력 적분 과정을 수식화하고 이를 유한 요소 해석 코드로 구현하였다. 그리고 양분법과 2점 뉴턴법을 채택하여 내연적 유한 요소 해석에서 효율적으로 사용될 수 있는 강건한 응력 적분 알고리즘을 제안하였다. 또한 소부 경화능을 묘사하기 위해 소부 경화 모델과 본 재료모델을 통합하였다. 단순한 문제에 본 모델을 적용하여 수행한 검증 해석 결과는 고전적인 반복법으로는 해를 얻을 수 없는 항복점 현상 문제에 본 모델이 신뢰할 수준으로 사용될 수 있지만 계산 시간은 증가할 수도 있다는 것을 보여주었다. 일축 실험을 재현할 수 있도록 결정된 재료 변수들을 이용하여 실험에 대응되는 수치해석이 수행되었다. 다양한 롤러 레벨링 조건에서의 V굽힘 해석은 실험적으로 관찰된 꺾임 결함과 그 감소 현상을 잘 보여주었다. 실험에서 확인된 소부 경화 거동은 소부 경화 단계를 포함하는 정적 덴트 해석에서 분석되었고, 소부 경화 효과가 수치 해석에서 전반적으로 묘사되었다. 결론적으로 본 연구에서 제안된 해석 방법은 다양한 금속 성형 공정에서 항복점 현상의 효과를 연구하는 것뿐만 아니라 꺾임 결함을 방지하기 위한 롤러 레벨링 조건을 추정하고 자동차 소부 경화 강판의 덴트 저항성을 예측하는데 유용하게 사용될 것으로 기대된다.1 Introduction 15 1.1 Yield-point phenomenon 15 1.2 Avoiding or utilizing the YPP 17 1.3 Literature review 20 1.3.1 YPP defect and its reduction method 20 1.3.2 Dent resistance considering BH behavior 21 1.3.3 Constitutive model and numerical analysis for YPP 21 1.3.4 Stress integration algorithms 23 1.3.5 Recent YPP studies 24 1.3.6 BH models 25 1.4 Objectives and outline 26 2 Experimental Investigations 29 2.1 Experimental observations for fluting defect and its reduction method with PLCS 30 2.1.1 Uniaxial tension test 30 2.1.2 Uniaxial cyclic test 32 2.1.3 Roller-leveling test 35 2.1.4 V-bending test 41 2.2 Experimental observations for static dent resistance considering BH behavior with 490DP 48 2.2.1 Uniaxial tension test with BH 48 2.2.2 Uniaxial tension-compression test with BH 55 2.2.3 Static dent test with BH 57 2.3 Summary 65 3 Material Modeling 67 3.1 Constitutive Model 68 3.1.1 YPP model 68 3.1.2 BH model 71 3.1.3 Integration of BH model into YPP model 73 3.2 Computational Implementation 74 3.2.1 BH calculation 74 3.2.2 Trial state assessment 77 3.2.3 Stress integration 78 3.2.4 Consistent tangent stiffness 87 3.2.5 Summary of the overall procedure 91 3.3 Summary 93 4 Validation of the Material Model 95 4.1 Single element analysis 96 4.2 Uniaxial tension and cyclic simulations 99 4.3 V-bending simulations 100 4.4 Cantilever bending simulation 103 4.5 Performance comparison 105 4.6 Single element BH simulations 107 4.7 Summary 110 5 Numerical Analysis 112 5.1 Numerical simulations of fluting defect and its reduction method for PLCS 113 5.1.1 Uniaxial tension and cyclic simulation 113 5.1.2 Roller-leveling and v-bending simulations 118 5.2 Numerical simulations of static dent resistance considering BH behavior for 490DP 130 5.2.1 Uniaxial tension and tension-compression simulations 130 5.2.2 Static dent simulations 138 5.3 Summary 146 6 Conclusion 151 A Appendix 155 A.1 Pseudocodes of UMAT subroutine for numerical simulations 155 A.2 Hertz contact problem for validating the parameters of the exponential pressure-overclosure relationship 159 References 162 Abstract (In Korean) 174Docto

Dynamics of Numerics & Spurious Behaviors in CFD Computations

The global nonlinear behavior of finite discretizations for constant time steps and fixed or adaptive grid spacings is studied using tools from dynamical systems theory. Detailed analysis of commonly used temporal and spatial discretizations for simple model problems is presented. The role of dynamics in the understanding of long time behavior of numerical integration and the nonlinear stability, convergence, and reliability of using time-marching approaches for obtaining steady-state numerical solutions in computational fluid dynamics (CFD) is explored. The study is complemented with examples of spurious behavior observed in steady and unsteady CFD computations. The CFD examples were chosen to illustrate non-apparent spurious behavior that was difficult to detect without extensive grid and temporal refinement studies and some knowledge from dynamical systems theory. Studies revealed the various possible dangers of misinterpreting numerical simulation of realistic complex flows that are constrained by available computing power. In large scale computations where the physics of the problem under study is not well understood and numerical simulations are the only viable means of solution, extreme care must be taken in both computation and interpretation of the numerical data. The goal of this paper is to explore the important role that dynamical systems theory can play in the understanding of the global nonlinear behavior of numerical algorithms and to aid the identification of the sources of numerical uncertainties in CFD

Numerical Solution of Optimal Control Problems with Explicit and Implicit Switches

This dissertation deals with the efficient numerical solution of switched optimal control problems whose dynamics may coincidentally be affected by both explicit and implicit switches. A framework is being developed for this purpose, in which both problem classes are uniformly converted into a mixed–integer optimal control problem with combinatorial constraints. Recent research results relate this problem class to a continuous optimal control problem with vanishing constraints, which in turn represents a considerable subclass of an optimal control problem with equilibrium constraints. In this thesis, this connection forms the foundation for a numerical treatment. We employ numerical algorithms that are based on a direct collocation approach and require, in particular, a highly accurate determination of the switching structure of the original problem. Due to the fact that the switching structure is a priori unknown in general, our approach aims to identify it successively. During this process, a sequence of nonlinear programs, which are derived by applying discretization schemes to optimal control problems, is solved approximatively. After each iteration, the discretization grid is updated according to the currently estimated switching structure. Besides a precise determination of the switching structure, it is of central importance to estimate the global error that occurs when optimal control problems are solved numerically. Again, we focus on certain direct collocation discretization schemes and analyze error contributions of individual discretization intervals. For this purpose, we exploit a relationship between discrete adjoints and the Lagrange multipliers associated with those nonlinear programs that arise from the collocation transcription process. This relationship can be derived with the help of a functional analytic framework and by interrelating collocation methods and Petrov–Galerkin finite element methods. In analogy to the dual-weighted residual methodology for Galerkin methods, which is well–known in the partial differential equation community, we then derive goal–oriented global error estimators. Based on those error estimators, we present mesh refinement strategies that allow for an equilibration and an efficient reduction of the global error. In doing so we note that the grid adaption processes with respect to both switching structure detection and global error reduction get along with each other. This allows us to distill an iterative solution framework. Usually, individual state and control components have the same polynomial degree if they originate from a collocation discretization scheme. Due to the special role which some control components have in the proposed solution framework it is desirable to allow varying polynomial degrees. This results in implementation problems, which can be solved by means of clever structure exploitation techniques and a suitable permutation of variables and equations. The resulting algorithm was developed in parallel to this work and implemented in a software package. The presented methods are implemented and evaluated on the basis of several benchmark problems. Furthermore, their applicability and efficiency is demonstrated. With regard to a future embedding of the described methods in an online optimal control context and the associated real-time requirements, an extension of the well–known multi–level iteration schemes is proposed. This approach is based on the trapezoidal rule and, compared to a full evaluation of the involved Jacobians, it significantly reduces the computational costs in case of sparse data matrices

The Asymptotic Safety Scenario in Quantum Gravity -- An Introduction

The asymptotic safety scenario in quantum gravity is reviewed, according to which a renormalizable quantum theory of the gravitational field is feasible which reconciles asymptotically safe couplings with unitarity. All presently known evidence is surveyed: (a) from the 2+\eps expansion, (b) from the perturbation theory of higher derivative gravity theories and a `large N' expansion in the number of matter fields, (c) from the 2-Killing vector reduction, and (d) from truncated flow equations for the effective average action. Special emphasis is given to the role of perturbation theory as a guide to `asymptotic safety'. Further it is argued that as a consequence of the scenario the selfinteractions appear two-dimensional in the extreme ultraviolet. Two appendices discuss the distinct roles of the ultraviolet renormalization in perturbation theory and in the flow equation formalism.Comment: 77p, 1 figure; v2: revised and updated; discussion of perturbation theory in higher derivative theories extended. To appear as topical review in CQ

A New Suite of Statistical Algorithms for Bayesian Model Fitting with Both Intrinsic and Extrinsic Uncertainties in Two Dimensions

Fitting a statistical model to data is one of the most important tools in any scientific or data-driven field, and rigorously fitting a two dimensional statistical model to data that has intrinsic uncertainties (error bars) in both the independent variable and the dependent variable is a daunting task, especially if the data also has extrinsic uncertainty (sample variance) that cannot be fully accounted for by the error bars. Here, I introduce a novel statistic (described as the Trotter, Reichart, Konz statistic, or TRK) developed in Trotter (2011) that is advantageous towards model-fitting in this "worst-case data" scenario, especially when compared to other methods. I implemented this statistic as a suite of fitting algorithms in C++ that comes equipped with many capabilities, including: support for any nonlinear model; probability distribution generation, correlation removal and custom priors for model parameters; asymmetric uncertainties in the data and/or model, and more. I also built an end-to-end website through which the algorithm can be used easily, but generally, with a high degree of customizability. The statistic is applicable to practically any data-driven field, and I show a few examples of its usage within the realm of astronomy. This thesis along with Trotter (2011) form the foundations for Trotter, Daniel E. Reichart, and Konz (2020), in preparation. The TRK source code and web-based calculator can be found at https://github.com/nickk124/TRK and https://skynet.unc.edu/rcr/calculator/trk, respectively.Bachelor of Scienc