Exponential integrators and a dual-scale model for wood drying

For the timber industry, the ability to simulate the drying of wood is invaluable for manufacturing high quality wood products. Mathematically, however, modelling the drying of a wet porous material, such as wood, is a diffcult task due to its heterogeneous and anisotropic nature, and the complex geometry of the underlying pore structure. The well{ developed macroscopic modelling approach involves writing down classical conservation equations at a length scale where physical quantities (e.g., porosity) can be interpreted as averaged values over a small volume (typically containing hundreds or thousands of pores). This averaging procedure produces balance equations that resemble those of a continuum with the exception that effective coeffcients appear in their deffnitions. Exponential integrators are numerical schemes for initial value problems involving a system of ordinary differential equations. These methods differ from popular Newton{Krylov implicit methods (i.e., those based on the backward differentiation formulae (BDF)) in that they do not require the solution of a system of nonlinear equations at each time step but rather they require computation of matrix{vector products involving the exponential of the Jacobian matrix. Although originally appearing in the 1960s, exponential integrators have recently experienced a resurgence in interest due to a greater undertaking of research in Krylov subspace methods for matrix function approximation. One of the simplest examples of an exponential integrator is the exponential Euler method (EEM), which requires, at each time step, approximation of φ(A)b, where φ(z) = (ez - 1)/z, A E Rnxn and b E Rn. For drying in porous media, the most comprehensive macroscopic formulation is TransPore [Perre and Turner, Chem. Eng. J., 86: 117-131, 2002], which features three coupled, nonlinear partial differential equations. The focus of the first part of this thesis is the use of the exponential Euler method (EEM) for performing the time integration of the macroscopic set of equations featured in TransPore. In particular, a new variable{ stepsize algorithm for EEM is presented within a Krylov subspace framework, which allows control of the error during the integration process. The performance of the new algorithm highlights the great potential of exponential integrators not only for drying applications but across all disciplines of transport phenomena. For example, when applied to well{ known benchmark problems involving single{phase liquid ow in heterogeneous soils, the proposed algorithm requires half the number of function evaluations than that required for an equivalent (sophisticated) Newton{Krylov BDF implementation. Furthermore for all drying configurations tested, the new algorithm always produces, in less computational time, a solution of higher accuracy than the existing backward Euler module featured in TransPore. Some new results relating to Krylov subspace approximation of '(A)b are also developed in this thesis. Most notably, an alternative derivation of the approximation error estimate of Hochbruck, Lubich and Selhofer [SIAM J. Sci. Comput., 19(5): 1552{1574, 1998] is provided, which reveals why it performs well in the error control procedure. Two of the main drawbacks of the macroscopic approach outlined above include the effective coefficients must be supplied to the model, and it fails for some drying configurations, where typical dual{scale mechanisms occur. In the second part of this thesis, a new dual{scale approach for simulating wood drying is proposed that couples the porous medium (macroscale) with the underlying pore structure (microscale). The proposed model is applied to the convective drying of softwood at low temperatures and is valid in the so{called hygroscopic range, where hygroscopically held liquid water is present in the solid phase and water exits only as vapour in the pores. Coupling between scales is achieved by imposing the macroscopic gradient on the microscopic field using suitably defined periodic boundary conditions, which allows the macroscopic ux to be defined as an average of the microscopic ux over the unit cell. This formulation provides a first step for moving from the macroscopic formulation featured in TransPore to a comprehensive dual{scale formulation capable of addressing any drying configuration. Simulation results reported for a sample of spruce highlight the potential and flexibility of the new dual{scale approach. In particular, for a given unit cell configuration it is not necessary to supply the effective coefficients prior to each simulation.

GPU accelerated algorithms for computing matrix function vector products with applications to exponential integrators and fractional diffusion

The efficient computation of matrix function vector products has become an important area of research in recent times, driven in particular by two important applications: the numerical solution of fractional partial differential equations and the integration of large systems of ordinary differential equations. In this work we consider a problem that combines these two applications, in the form of a numerical solution algorithm for fractional reaction diffusion equations that after spatial discretisation, is advanced in time using the exponential Euler method. We focus on the efficient implementation of the algorithm on Graphics Processing Units (GPU), as we wish to make use of the increased computational power available with this hardware. We compute the matrix function vector products using the contour integration method in [N. Hale, N. Higham, and L. Trefethen. Computing Aα, log(A), and related matrix functions by contour integrals. SIAM J. Numer. Anal., 46(5):2505–2523, 2008]. Multiple levels of preconditioning are applied to reduce the GPU memory footprint and to further accelerate convergence. We also derive an error bound for the convergence of the contour integral method that allows us to pre-determine the appropriate number of quadrature points. Results are presented that demonstrate the effectiveness of the method for large two-dimensional problems, showing a speedup of more than an order of magnitude compared to a CPU-only implementation.

Efficient simulation of unsaturated flow using exponential time integration

We assess the performance of an exponential integrator for advancing stiff, semidiscrete formulations of the unsaturated Richards equation in time. The scheme is of second order and explicit in nature but requires the action of the matrix function φ(A) where φ(z) = [exp(z) - 1]/z on a suitability defined vector v at each time step. When the matrix A is large and sparse, φ(A)v can be approximated by Krylov subspace methods that require only matrix-vector products with A. We prove that despite the use of this approximation the scheme remains second order. Furthermore, we provide a practical variable-stepsize implementation of the integrator by deriving an estimate of the local error that requires only a single additional function evaluation. Numerical experiments performed on two-dimensional test problems demonstrate that this implementation outperforms second-order, variable-stepsize implementations of the backward differentiation formulae.

Digital signal processing

Signal Processing (SP) is a subject of central importance in engineering and the applied sciences. Signals are information-bearing functions, and SP deals with the analysis and processing of signals (by dedicated systems) to extract or modify information. Signal processing is necessary because signals normally contain information that is not readily usable or understandable, or which might be disturbed by unwanted sources such as noise. Although many signals are non-electrical, it is common to convert them into electrical signals for processing. Most natural signals (such as acoustic and biomedical signals) are continuous functions of time, with these signals being referred to as analog signals. Prior to the onset of digital computers, Analog Signal Processing (ASP) and analog systems were the only tool to deal with analog signals. Although ASP and analog systems are still widely used, Digital Signal Processing (DSP) and digital systems are attracting more attention, due in large part to the significant advantages of digital systems over the analog counterparts. These advantages include superiority in performance,s peed, reliability, efficiency of storage, size and cost. In addition, DSP can solve problems that cannot be solved using ASP, like the spectral analysis of multicomonent signals, adaptive filtering, and operations at very low frequencies. Following the recent developments in engineering which occurred in the 1980's and 1990's, DSP became one of the world's fastest growing industries. Since that time DSP has not only impacted on traditional areas of electrical engineering, but has had far reaching effects on other domains that deal with information such as economics, meteorology, seismology, bioengineering, oceanology, communications, astronomy, radar engineering, control engineering and various other applications. This book is based on the Lecture Notes of Associate Professor Zahir M. Hussain at RMIT University (Melbourne, 2001-2009), the research of Dr. Amin Z. Sadik (at QUT & RMIT, 2005-2008), and the Note of Professor Peter O'Shea at Queensland University of Technology. Part I of the book addresses the representation of analog and digital signals and systems in the time domain and in the frequency domain. The core topics covered are convolution, transforms (Fourier, Laplace, Z. Discrete-time Fourier, and Discrete Fourier), filters, and random signal analysis. There is also a treatment of some important applications of DSP, including signal detection in noise, radar range estimation, banking and financial applications, and audio effects production. Design and implementation of digital systems (such as integrators, differentiators, resonators and oscillators are also considered, along with the design of conventional digital filters. Part I is suitable for an elementary course in DSP. Part II (which is suitable for an advanced signal processing course), considers selected signal processing systems and techniques. Core topics covered are the Hilbert transformer, binary signal transmission, phase-locked loops, sigma-delta modulation, noise shaping, quantization, adaptive filters, and non-stationary signal analysis. Part III presents some selected advanced DSP topics. We hope that this book will contribute to the advancement of engineering education and that it will serve as a general reference book on digital signal processing.

Krylov subspace approximations for the exponential Euler method: Error estimates and the harmonic Ritz approximant

We study Krylov subspace methods for approximating the matrix-function vector product φ(tA)b where φ(z) = [exp(z) - 1]/z. This product arises in the numerical integration of large stiff systems of differential equations by the Exponential Euler Method, where A is the Jacobian matrix of the system. Recently, this method has found application in the simulation of transport phenomena in porous media within mathematical models of wood drying and groundwater flow. We develop an a posteriori upper bound on the Krylov subspace approximation error and provide a new interpretation of a previously published error estimate. This leads to an alternative Krylov approximation to φ(tA)b, the so-called Harmonic Ritz approximant, which we find does not exhibit oscillatory behaviour of the residual error.

The rise of co-creative consumers : user experience sharing behaviour in online communities

This thesis examines consumer initiated value co-creation behaviour in the context of convergent mobile online services using a Service-Dominant logic (SD logic) theoretical framework. It focuses on non-reciprocal marketing phenomena such as open innovation and user generated content whereby new viable business models are derived and consumer roles and community become essential to the success of business. Attention to customers. roles and personalised experiences in value co-creation has been recognised in the literature (e.g., Prahalad & Ramaswamy, 2000; Prahalad, 2004; Prahalad & Ramaswamy, 2004). Similarly, in a subsequent iteration of their 2004 version of the foundations of SD logic, Vargo and Lusch (2006) replaced the concept of value co-production with value co-creation and suggested that a value co-creation mindset is essential to underpin the firm-customer value creation relationship. Much of this focus, however, has been limited to firm initiated value co-creation (e.g., B2B or B2C), while consumer initiated value creation, particularly consumer-to-consumer (C2C) has received little attention in the SD logic literature. While it is recognised that not every consumer wishes to make the effort to engage extensively in co-creation processes (MacDonald & Uncles, 2009), some consumers may not be satisfied with a standard product, instead they engage in the effort required for personalisation that potentially leads to greater value for themselves, and which may benefit not only the firm, but other consumers as well. Literature suggests that there are consumers who do, and as a result initiate such behaviour and expend effort to engage in co-creation activity (e.g., Gruen, Osmonbekov and Czaplewski, 2006; 2007 MacDonald & Uncles, 2009). In terms of consumers. engagement in value proposition (co-production) and value actualisation (co-creation), SD logic (Vargo & Lusch, 2004, 2008) provides a new lens that enables marketing scholars to transcend existing marketing theory and facilitates marketing practitioners to initiate service centric and value co-creation oriented marketing practices. Although the active role of the consumer is acknowledged in the SD logic oriented literature, we know little about how and why consumers participate in a value co-creation process (Payne, Storbacka, & Frow, 2008). Literature suggests that researchers should focus on areas such as C2C interaction (Gummesson 2007; Nicholls 2010) and consumer experience sharing and co-creation (Belk 2009; Prahalad & Ramaswamy 2004). In particular, this thesis seeks to better understand consumer initiated value co-creation, which is aligned with the notion that consumers can be resource integrators (Baron & Harris, 2008) and more. The reason for this focus is that consumers today are more empowered in both online and offline contexts (Füller, Mühlbacher, Matzler, & Jawecki, 2009; Sweeney, 2007). Active consumers take initiatives to engage and co-create solutions with other active actors in the market for their betterment of life (Ballantyne & Varey, 2006; Grönroos & Ravald, 2009). In terms of the organisation of the thesis, this thesis first takes a „zoom-out. (Vargo & Lusch, 2011) approach and develops the Experience Co-Creation (ECo) framework that is aligned with balanced centricity (Gummesson, 2008) and Actor-to-Actor worldview (Vargo & Lusch, 2011). This ECo framework is based on an extended „SD logic friendly lexicon. (Lusch & Vargo, 2006): value initiation and value initiator, value-in-experience, betterment centricity and betterment outcomes, and experience co-creation contexts derived from five gaps identified from the SD logic literature review. The framework is also designed to accommodate broader marketing phenomena (i.e., both reciprocal and non-reciprocal marketing phenomena). After zooming out and establishing the ECo framework, the thesis takes a zoom-in approach and places attention back on the value co-creation process. Owing to the scope of the current research, this thesis focuses specifically on non-reciprocal value co-creation phenomena initiated by consumers in online communities. Two emergent concepts: User Experience Sharing (UES) and Co-Creative Consumers are proposed grounded in the ECo framework. Together, these two theorised concepts shed light on the following two propositions: (1) User Experience Sharing derives value-in-experience as consumers make initiative efforts to participate in value co-creation, and (2) Co-Creative Consumers are value initiators who perform UES. Three research questions were identified underpinning the scope of this research: RQ1: What factors influence consumers to exhibit User Experience Sharing behaviour? RQ2: Why do Co-Creative Consumers participate in User Experience Sharing as part of value co-creation behaviour? RQ3: What are the characteristics of Co-Creative Consumers? To answer these research questions, two theoretical models were developed: the User Experience Sharing Behaviour Model (UESBM) grounded in the Theory of Planned Behaviour framework, and the Co-Creative Consumer Motivation Model (CCMM) grounded in the Motivation, Opportunity, Ability framework. The models use SD logic consistent constructs and draw upon multiple streams of literature including consumer education, consumer psychology and consumer behaviour, and organisational psychology and organisational behaviour. These constructs include User Experience Sharing with Other Consumers (UESC), User Experience Sharing with Firms (UESF), Enjoyment in Helping Others (EIHO), Consumer Empowerment (EMP), Consumer Competence (COMP), and Intention to Engage in User Experience Sharing (INT), Attitudes toward User Experience Sharing (ATT) and Subjective Norm (SN) in the UESBM, and User Experience Sharing (UES), Consumer Citizenship (CIT), Relating Needs of Self (RELS) and Relating Needs of Others (RELO), Newness (NEW), Mavenism (MAV), Use Innovativeness (UI), Personal Initiative (PIN) and Communality (COMU) in the CCMM. Many of these constructs are relatively new to marketing and require further empirical evidence for support. Two studies were conducted to underpin the corresponding research questions. Study One was conducted to calibrate and re-specify the proposed models. Study Two was a replica study to confirm the proposed models. In Study One, data were collected from a PC DIY online community. In Study Two, a majority of data were collected from Apple product online communities. The data were examined using structural equation modelling and cluster analysis. Considering the nature of the forums, the Study One data is considered to reflect some characteristics of Prosumers and the Study Two data is considered to reflect some characteristics of Innovators. The results drawn from two independent samples (N = 326 and N = 294) provide empirical support for the overall structure theorised in the research models. The results in both models show that Enjoyment in Helping Others and Consumer Competence in the UESBM, and Consumer Citizenship and Relating Needs in CCMM have significant impacts on UES. The consistent results appeared in both Study One and Study Two. The results also support the conceptualisation of Co-Creative Consumers and indicate Co-Creative Consumers are individuals who are able to relate the needs of themselves and others and feel a responsibility to share their valuable personal experiences. In general, the results shed light on "How and why consumers voluntarily participate in the value co-creation process?. The findings provide evidence to conceptualise User Experience Sharing behaviour as well as the Co-Creative Consumer using the lens of SD logic. This research is a pioneering study that incorporates and empirically tests SD logic consistent constructs to examine a particular area of the logic – that is consumer initiated value co-creation behaviour. This thesis also informs practitioners about how to facilitate and understand factors that engage with either firm or consumer initiated online communities.

Low rank Runge-Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise

In this paper we extend the ideas of Brugnano, Iavernaro and Trigiante in their development of HBVM($s,r$) methods to construct symplectic Runge-Kutta methods for all values of $s$ and $r$ with $s\geq r$. However, these methods do not see the dramatic performance improvement that HBVMs can attain. Nevertheless, in the case of additive stochastic Hamiltonian problems an extension of these ideas, which requires the simulation of an independent Wiener process at each stage of a Runge-Kutta method, leads to methods that have very favourable properties. These ideas are illustrated by some simple numerical tests for the modified midpoint rule.

Performance analysis of PTP components for IEC 61850 process bus applications

New substation automation applications, such as sampled value process buses and synchrophasors, require sampling accuracy of 1 µs or better. The Precision Time Protocol (PTP), IEEE Std 1588, achieves this level of performance and integrates well into Ethernet based substation networks. This paper takes a systematic approach to the performance evaluation of commercially available PTP devices (grandmaster, slave, transparent and boundary clocks) from a variety of manufacturers. The ``error budget'' is set by the performance requirements of each application. The ``expenditure'' of this error budget by each component is valuable information for a system designer. The component information is used to design a synchronization system that meets the overall functional requirements. The quantitative performance data presented shows that this testing is effective and informative. Results from testing PTP performance in the presence of sampled value process bus traffic demonstrate the benefit of a ``bottom up'' component testing approach combined with ``top down'' system verification tests. A test method that uses a precision Ethernet capture card, rather than dedicated PTP test sets, to determine the Correction Field Error of transparent clocks is presented. This test is particularly relevant for highly loaded Ethernet networks with stringent timing requirements. The methods presented can be used for development purposes by manufacturers, or by system integrators for acceptance testing. A sampled value process bus was used as the test application for the systematic approach described in this paper. The test approach was applied, components were selected, and the system performance verified to meet the application's requirements. Systematic testing, as presented in this paper, is applicable to a range of industries that use, rather than develop, PTP for time transfer.

A dual-scale modelling approach for drying hygroscopic porous media

A new dualscale modelling approach is presented for simulating the drying of a wet hygroscopic porous material that couples the porous medium (macroscale) with the underlying pore structure (microscale). The proposed model is applied to the convective drying of wood at low temperatures and is valid in the so-called hygroscopic range, where hygroscopically held liquid water is present in the solid phase and water exits only as vapour in the pores. Coupling between scales is achieved by imposing the macroscopic gradients of moisture content and temperature on the microscopic field using suitably-defined periodic boundary conditions, which allows the macroscopic mass and thermal fluxes to be defined as averages of the microscopic fluxes over the unit cell. This novel formulation accounts for the intricate coupling of heat and mass transfer at the microscopic scale but reduces to a classical homogenisation approach if a linear relationship is assumed between the microscopic gradient and flux. Simulation results for a sample of spruce wood highlight the potential and flexibility of the new dual-scale approach. In particular, for a given unit cell configuration it is not necessary to propose the form of the macroscopic fluxes prior to the simulations because these are determined as a direct result of the dual-scale formulation.

A variable-stepsize Jacobian-free exponential integrator for simulating transport in heterogeneous porous media : application to wood drying

A Jacobian-free variable-stepsize method is developed for the numerical integration of the large, stiff systems of differential equations encountered when simulating transport in heterogeneous porous media. Our method utilises the exponential Rosenbrock-Euler method, which is explicit in nature and requires a matrix-vector product involving the exponential of the Jacobian matrix at each step of the integration process. These products can be approximated using Krylov subspace methods, which permit a large integration stepsize to be utilised without having to precondition the iterations. This means that our method is truly "Jacobian-free" - the Jacobian need never be formed or factored during the simulation. We assess the performance of the new algorithm for simulating the drying of softwood. Numerical experiments conducted for both low and high temperature drying demonstrates that the new approach outperforms (in terms of accuracy and efficiency) existing simulation codes that utilise the backward Euler method via a preconditioned Newton-Krylov strategy.

Performance assessment of exponential Rosenbrock method for large systems of ODE

This paper studies time integration methods for large stiff systems of ordinary differential equations (ODEs) of the form u'(t) = g(u(t)). For such problems, implicit methods generally outperform explicit methods, since the time step is usually less restricted by stability constraints. Recently, however, explicit so-called exponential integrators have become popular for stiff problems due to their favourable stability properties. These methods use matrix-vector products involving exponential-like functions of the Jacobian matrix, which can be approximated using Krylov subspace methods that require only matrix-vector products with the Jacobian. In this paper, we implement exponential integrators of second, third and fourth order and demonstrate that they are competitive with well-established approaches based on the backward differentiation formulas and a preconditioned Newton-Krylov solution strategy.

Structure-preserving Runge-Kutta methods for stochastic Hamiltonian equations with additive noise

There has been considerable recent work on the development of energy conserving one-step methods that are not symplectic. Here we extend these ideas to stochastic Hamiltonian problems with additive noise and show that there are classes of Runge-Kutta methods that are very effective in preserving the expectation of the Hamiltonian, but care has to be taken in how the Wiener increments are sampled at each timestep. Some numerical simulations illustrate the performance of these methods.

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.

A gray-box neural network-based model identification and fault estimation scheme for nonlinear dynamic systems

A novel gray-box neural network model (GBNNM), including multi-layer perception (MLP) neural network (NN) and integrators, is proposed for a model identification and fault estimation (MIFE) scheme. With the GBNNM, both the nonlinearity and dynamics of a class of nonlinear dynamic systems can be approximated. Unlike previous NN-based model identification methods, the GBNNM directly inherits system dynamics and separately models system nonlinearities. This model corresponds well with the object system and is easy to build. The GBNNM is embedded online as a normal model reference to obtain the quantitative residual between the object system output and the GBNNM output. This residual can accurately indicate the fault offset value, so it is suitable for differing fault severities. To further estimate the fault parameters (FPs), an improved extended state observer (ESO) using the same NNs (IESONN) from the GBNNM is proposed to avoid requiring the knowledge of ESO nonlinearity. Then, the proposed MIFE scheme is applied for reaction wheels (RW) in a satellite attitude control system (SACS). The scheme using the GBNNM is compared with other NNs in the same fault scenario, and several partial loss of effect (LOE) faults with different severities are considered to validate the effectiveness of the FP estimation and its superiority.