What do we need in programming languages for mathematical software? /

"UIUCDCS-R-74-652"

RcppArmadillo : accelerating R with high-performance C++ linear algebra

The R statistical environment and language has demonstrated particular strengths for interactive development of statistical algorithms, as well as data modelling and visualisation. Its current implementation has an interpreter at its core which may result in a performance penalty in comparison to directly executing user algorithms in the native machine code of the host CPU. In contrast, the C++ language has no built-in visualisation capabilities, handling of linear algebra or even basic statistical algorithms; however, user programs are converted to high-performance machine code, ahead of execution. A new method avoids possible speed penalties in R by using the Rcpp extension package in conjunction with the Armadillo C++ matrix library. In addition to the inherent performance advantages of compiled code, Armadillo provides an easy-to-use template-based meta-programming framework, allowing the automatic pooling of several linear algebra operations into one, which in turn can lead to further speedups. With the aid of Rcpp and Armadillo, conversion of linear algebra centered algorithms from R to C++ becomes straightforward. The algorithms retains the overall structure as well as readability, all while maintaining a bidirectional link with the host R environment. Empirical timing comparisons of R and C++ implementations of a Kalman filtering algorithm indicate a speedup of several orders of magnitude.

Some practical issues in the design and analysis of computer experiments

Deterministic computer simulations of physical experiments are now common techniques in science and engineering. Often, physical experiments are too time consuming, expensive or impossible to conduct. Complex computer models or codes, rather than physical experiments lead to the study of computer experiments, which are used to investigate many scientific phenomena of this nature. A computer experiment consists of a number of runs of the computer code with different input choices. The Design and Analysis of Computer Experiments is a rapidly growing technique in statistical experimental design. This thesis investigates some practical issues in the design and analysis of computer experiments and attempts to answer some of the questions faced by experimenters using computer experiments. In particular, the question of the number of computer experiments and how they should be augmented is studied and attention is given to when the response is a function over time.

Towards a model of spray-canopy interactions : interception, shatter, bounce and retention of droplets on horizontal leaves

Pesticides used in agricultural systems must be applied in economically viable and environmentally sensitive ways, and this often requires expensive field trials on spray deposition and retention by plant foliage. Computational models to describe whether a spray droplet sticks (adheres), bounces or shatters on impact, and if any rebounding parent or shatter daughter droplets are recaptured, would provide an estimate of spray retention and thereby act as a useful guide prior to any field trials. Parameter-driven interactive software has been implemented to enable the end-user to study and visualise droplet interception and impaction on a single, horizontal leaf. Living chenopodium, wheat and cotton leaves have been scanned to capture the surface topography and realistic virtual leaf surface models have been generated. Individual leaf models have then been subjected to virtual spray droplets and predictions made of droplet interception with the virtual plant leaf. Thereafter, the impaction behaviour of the droplets and the subsequent behaviour of any daughter droplets, up until re-capture, are simulated to give the predicted total spray retention by the leaf. A series of critical thresholds for the stick, bounce, and shatter elements in the impaction process have been developed for different combinations of formulation, droplet size and velocity, and leaf surface characteristics to provide this output. The results show that droplet properties, spray formulations and leaf surface characteristics all influence the predicted amount of spray retained on a horizontal leaf surface. Overall the predicted spray retention increases as formulation surface tension, static contact angle, droplet size and velocity decreases. Predicted retention on cotton is much higher than on chenopodium. The average predicted retention on a single horizontal leaf across all droplet size, velocity and formulations scenarios tested, is 18, 30 and 85% for chenopodium, wheat and cotton, respectively.

Two-scale computational modelling of unsaturated flow in soils exhibiting small scale heterogeneities

Unsaturated water flow in soil is commonly modelled using Richards’ equation, which requires the hydraulic properties of the soil (e.g., porosity, hydraulic conductivity, etc.) to be characterised. Naturally occurring soils, however, are heterogeneous in nature, that is, they are composed of a number of interwoven homogeneous soils each with their own set of hydraulic properties. When the length scale of these soil heterogeneities is small, numerical solution of Richards’ equation is computationally impractical due to the immense effort and refinement required to mesh the actual heterogeneous geometry. A classic way forward is to use a macroscopic model, where the heterogeneous medium is replaced with a fictitious homogeneous medium, which attempts to give the average flow behaviour at the macroscopic scale (i.e., at a scale much larger than the scale of the heterogeneities). Using the homogenisation theory, a macroscopic equation can be derived that takes the form of Richards’ equation with effective parameters. A disadvantage of the macroscopic approach, however, is that it fails in cases when the assumption of local equilibrium does not hold. This limitation has seen the introduction of two-scale models that include at each point in the macroscopic domain an additional flow equation at the scale of the heterogeneities (microscopic scale). This report outlines a well-known two-scale model and contributes to the literature a number of important advances in its numerical implementation. These include the use of an unstructured control volume finite element method and image-based meshing techniques, that allow for irregular micro-scale geometries to be treated, and the use of an exponential time integration scheme that permits both scales to be resolved simultaneously in a completely coupled manner. Numerical comparisons against a classical macroscopic model confirm that only the two-scale model correctly captures the important features of the flow for a range of parameter values.

Towards a Modeling Environment for Earth Systems Simulations

The developments of models in Earth Sciences, e.g. for earthquake prediction and for the simulation of mantel convection, are fare from being finalized. Therefore there is a need for a modelling environment that allows scientist to implement and test new models in an easy but flexible way. After been verified, the models should be easy to apply within its scope, typically by setting input parameters through a GUI or web services. It should be possible to link certain parameters to external data sources, such as databases and other simulation codes. Moreover, as typically large-scale meshes have to be used to achieve appropriate resolutions, the computational efficiency of the underlying numerical methods is important. Conceptional this leads to a software system with three major layers: the application layer, the mathematical layer, and the numerical algorithm layer. The latter is implemented as a C/C++ library to solve a basic, computational intensive linear problem, such as a linear partial differential equation. The mathematical layer allows the model developer to define his model and to implement high level solution algorithms (e.g. Newton-Raphson scheme, Crank-Nicholson scheme) or choose these algorithms form an algorithm library. The kernels of the model are generic, typically linear, solvers provided through the numerical algorithm layer. Finally, to provide an easy-to-use application environment, a web interface is (semi-automatically) built to edit the XML input file for the modelling code. In the talk, we will discuss the advantages and disadvantages of this concept in more details. We will also present the modelling environment escript which is a prototype implementation toward such a software system in Python (see www.python.org). Key components of escript are the Data class and the PDE class. Objects of the Data class allow generating, holding, accessing, and manipulating data, in such a way that the actual, in the particular context best, representation is transparent to the user. They are also the key to establish connections with external data sources. PDE class objects are describing (linear) partial differential equation objects to be solved by a numerical library. The current implementation of escript has been linked to the finite element code Finley to solve general linear partial differential equations. We will give a few simple examples which will illustrate the usage escript. Moreover, we show the usage of escript together with Finley for the modelling of interacting fault systems and for the simulation of mantel convection.

Unified form language: A domain-specific language for weak formulations of partial differential equations

We present the Unified Form Language (UFL), which is a domain-specific language for representing weak formulations of partial differential equations with a view to numerical approximation. Features of UFL include support for variational forms and functionals, automatic differentiation of forms and expressions, arbitrary function space hierarchies formultifield problems, general differential operators and flexible tensor algebra. With these features, UFL has been used to effortlessly express finite element methods for complex systems of partial differential equations in near-mathematical notation, resulting in compact, intuitive and readable programs. We present in this work the language and its construction. An implementation of UFL is freely available as an open-source software library. The library generates abstract syntax tree representations of variational problems, which are used by other software libraries to generate concrete low-level implementations. Some application examples are presented and libraries that support UFL are highlighted. © 2014 ACM.

Integrated magnetic components for high-power high-current dc-dc converters

This thesis is focused on the design and development of an integrated magnetic (IM) structure for use in high-power high-current power converters employed in renewable energy applications. These applications require low-cost, high efficiency and high-power density magnetic components and the use of IM structures can help achieve this goal. A novel CCTT-core split-winding integrated magnetic (CCTT IM) is presented in this thesis. This IM is optimized for use in high-power dc-dc converters. The CCTT IM design is an evolution of the traditional EE-core integrated magnetic (EE IM). The CCTT IM structure uses a split-winding configuration allowing for the reduction of external leakage inductance, which is a problem for many traditional IM designs, such as the EE IM. Magnetic poles are incorporated to help shape and contain the leakage flux within the core window. These magnetic poles have the added benefit of minimizing the winding power loss due to the airgap fringing flux as they shape the fringing flux away from the split-windings. A CCTT IM reluctance model is developed which uses fringing equations to accurately predict the most probable regions of fringing flux around the pole and winding sections of the device. This helps in the development of a more accurate model as it predicts the dc and ac inductance of the component. A CCTT IM design algorithm is developed which relies heavily on the reluctance model of the CCTT IM. The design algorithm is implemented using the mathematical software tool Mathematica. This algorithm is modular in structure and allows for the quick and easy design and prototyping of the CCTT IM. The algorithm allows for the investigation of the CCTT IM boxed volume with the variation of input current ripple, for different power ranges, magnetic materials and frequencies. A high-power 72 kW CCTT IM prototype is designed and developed for use in an automotive fuelcell-based drivetrain. The CCTT IM design algorithm is initially used to design the component while 3D and 2D finite element analysis (FEA) software is used to optimize the design. Low-cost and low-power loss ferrite 3C92 is used for its construction, and when combined with a low number of turns results in a very efficient design. A paper analysis is undertaken which compares the performance of the high-power CCTT IM design with that of two discrete inductors used in a two-phase (2L) interleaved converter. The 2L option consists of two discrete inductors constructed from high dc-bias material. Both topologies are designed for the same worst-case phase current ripple conditions and this ensures a like-for-like comparison. The comparison indicates that the total magnetic component boxed volume of both converters is similar while the CCTT IM has significantly lower power loss. Experimental results for the 72 kW, (155 V dc, 465 A dc input, 420 V dc output) prototype validate the CCTT IM concept where the component is shown to be 99.7 % efficient. The high-power experimental testing was conducted at General Motors advanced technology center in Torrence, Los Angeles. Calorific testing was used to determine the power loss in the CCTT IM component. Experimental 3.8 kW results and a 3.8 kW prototype compare and contrast the ferrite CCTT IM and high dc-bias 2L concepts over the typical operating range of a fuelcell under like-for-like conditions. The CCTT IM is shown to perform better than the 2L option over the entire power range. An 8 kW ferrite CCTT IM prototype is developed for use in photovoltaic (PV) applications. The CCTT IM is used in a boost pre-regulator as part of the PV power stage. The CCTT IM is compared with an industry standard 2L converter consisting of two discrete ferrite toroidal inductors. The magnetic components are compared for the same worst-case phase current ripple and the experimental testing is conducted over the operation of a PV panel. The prototype CCTT IM allows for a 50 % reduction in total boxed volume and mass in comparison to the baseline 2L option, while showing increased efficiency.

Simulation of dynamic pinion course using Runge-Kutta's method and impact modeling

Once defined the relationship between the Starter Motor components and their functions, it is possible to develop a mathematical model capable to predict the Starter behavior during operation. One important aspect is the engagement system behavior. The development of a mathematical tool capable of predicting it is a valuable step in order to reduce the design time, cost and engineering efforts. A mathematical model, represented by differential equations, can be developed using physics laws, evaluating force balance and energy flow through the systems degrees of freedom. Another important physical aspect to be considered in this modeling is the impact conditions (particularly on the pinion and ring-gear contact). This work is a report of those equations application on available mathematical software and the resolution of those equations by Runge-Kutta's numerical integration method, in order to build an accessible engineering tool. Copyright © 2011 SAE International.