On Asymptotic Global Error Estimation and Control of Finite Difference Solutions for Semilinear Parabolic Equations

The aim of this paper is to extend the global error estimation and control addressed in Lang and Verwer [SIAM J. Sci. Comput. 29, 2007] for initial value problems to finite difference solutions of semilinear parabolic partial differential equations. The approach presented there is combined with an estimation of the PDE spatial truncation error by Richardson extrapolation to estimate the overall error in the computed solution. Approximations of the error transport equations for spatial and temporal global errors are derived by using asymptotic estimates that neglect higher order error terms for sufficiently small step sizes in space and time. Asymptotic control in a discrete $L_2$-norm is achieved through tolerance proportionality and uniform or adaptive mesh refinement. Numerical examples are used to illustrate the reliability of the estimation and control strategies

Evolutionary algorithms for the modeling of bioreactors

This work aims to study the application of Genetic Algorithms in anaerobic digestion modeling, in particular when using dynamical models. Along the work, different types of bioreactors are shown, such as batch, semi-batch and continuous, as well as their mathematical modeling. The work intendeds to estimate the parameter values of two biological reaction model. For that, simulated results, where only one output variable, the produced biogas, is known, are fitted to the model results. For this reason, the problems associated with reverse optimization are studied, using some graphics that provide clues to the sensitivity and identifiability associated with the problem. Particular solutions obtained by the identifiability analysis using GENSSI and DAISY softwares are also presented. Finally, the optimization is performed using genetic algorithms. During this optimization the need to improve the convergence of genetic algorithms was felt. This need has led to the development of an adaptation of the genetic algorithms, which we called Neighbored Genetic Algorithms (NGA1 and NGA2). In order to understand if this new approach overcomes the Basic Genetic Algorithms (BGA) and achieves the proposed goals, a study of 100 full optimization runs for each situation was further developed. Results show that NGA1 and NGA2 are statistically better than BGA. However, because it was not possible to obtain consistent results, the Nealder-Mead method was used, where the initial guesses were the estimated results from GA; Algoritmos Evolucionários para a Modelação de Bioreactores Resumo: Neste trabalho procura-se estudar os algoritmos genéticos com aplicação na modelação da digestão anaeróbia e, em particular, quando se utilizam modelos dinâmicos. Ao longo do mesmo, são apresentados diferentes tipos de bioreactores, como os batch, semi-batch e contínuos, bem como a modelação matemática dos mesmos. Neste trabalho procurou-se estimar o valor dos parâmetros que constam num modelo de digestão anaeróbia para o ajustar a uma situação simulada onde apenas se conhece uma variável de output, o biogas produzido. São ainda estudados os problemas associados à optimização inversa com recurso a alguns gráficos que fornecem pistas sobre a sensibilidade e identifiacabilidade associadas ao problema da modelação da digestão anaeróbia. São ainda apresentadas soluções particulares de idenficabilidade obtidas através dos softwares GENSSI e DAISY. Finalmente é realizada a optimização do modelo com recurso aos algoritmos genéticos. No decorrer dessa optimização sentiu-se a necessidade de melhorar a convergência e, portanto, desenvolveu-se ainda uma adaptação dos algoritmos genéticos a que se deu o nome de Neighboured Genetic Algorithms (NGA1 e NGA2). No sentido de se compreender se as adaptações permitiam superar os algoritmos genéticos básicos e atingir as metas propostas, foi ainda desenvolvido um estudo em que o processo de optimização foi realizado 100 vezes para cada um dos métodos, o que permitiu concluir, estatisticamente, que os BGA foram superados pelos NGA1 e NGA2. Ainda assim, porque não foi possivel obter consistência nos resultados, foi usado o método de Nealder-Mead utilizado como estimativa inicial os resultados obtidos pelos algoritmos genéticos

Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review

A review of diagonally implicit Runge-Kutta (DIRK) methods applied to rst-order ordinary di erential equations (ODEs) is undertaken. The goal of this review is to summarize the characteristics, assess the potential, and then design several nearly optimal, general purpose, DIRK-type methods. Over 20 important aspects of DIRKtype methods are reviewed. A design study is then conducted on DIRK-type methods having from two to seven implicit stages. From this, 15 schemes are selected for general purpose application. Testing of the 15 chosen methods is done on three singular perturbation problems. Based on the review of method characteristics, these methods focus on having a stage order of two, sti accuracy, L-stability, high quality embedded and dense-output methods, small magnitudes of the algebraic stability matrix eigenvalues, small values of aii, and small or vanishing values of the internal stability function for large eigenvalues of the Jacobian. Among the 15 new methods, ESDIRK4(3)6L[2]SA is recommended as a good default method for solving sti problems at moderate error tolerances

An Adaptive Method for Calculating Blow-Up Solutions

Reactive-diffusive systems modeling physical phenomena in certain situations develop a singularity at a finite value of the independent variable referred to as blow-up. The attempt to find the blow-up time analytically is most often impossible, thus requiring a numerical determination of the value. The numerical methods often use a priori knowledge of the blow-up solution such as monotonicity or self-similarity. For equations where such a priori knowledge is unavailable, ad hoc methods were constructed. The object of this research is to develop a simple and consistent approach to find numerically the blow-up solution without having a priori knowledge or resorting to other ad hoc methods. The proposed method allows the investigator the ability to distinguish whether a singular solution or a non-singular solution exists on a given interval. Step size in the vicinity of a singular solution is automatically adjusted. The programming of the proposed method is simple and uses well-developed software for most of the auxiliary routines. The proposed numerical method is mainly concerned with the integration of nonlinear integral equations with Abel-type kernels developed from combustion problems, but may be used on similar equations from other fields. To demonstrate the flexibility of the proposed method, it is applied to ordinary differential equations with blow-up solutions or to ordinary differential equations which exhibit extremely stiff structure

Adaptive time-integration for goal-oriented and coupled problems

We consider efficient methods for the partitioned time-integration of multiphysics problems, which commonly exhibit a multiscale behavior, requiring independent time-grids. Examples are fluid structure interaction in e.g., the simulation of blood-flow or cooling of rocket engines, or ocean-atmosphere-vegetation interaction. The ideal method for solving these problems allows independent and adaptive time-grids, higher order time-discretizations, is fast and robust, and allows the coupling of existing subsolvers, executed in parallel. We consider Waveform relaxation (WR) methods, which can have all of these properties. WR methods iterate on continuous-in-time interface functions, obtained via suitable interpolation. The difficulty is to find suitable convergence acceleration, which is required for the iteration converge quickly. We develop a fast and highly robust, second order in time, adaptive WR method for unsteady thermal fluid structure interaction (FSI), modelled by heterogeneous coupled linear heat equations. We use a Dirichlet-Neumann coupling at the interface and an analytical optimal relaxation parameter derived for the fully-discrete scheme. While this method is sequential, it is notably faster and more robust than similar parallel methods.We further develop a novel, parallel WR method, using asynchronous communication techniques during time-integration to accelerate convergence. Instead of exchanging interpolated time-dependent functions at the end of each time-window or iteration, we exchange time-point data immediately after each timestep. The analytical description and convergence results of this method generalize existing WR theory.Since WR methods allow coupling of problems in a relative black-box manner, we developed adapters to PDE-subsolvers implemented using DUNE and FEniCS. We demonstrate this coupling in a thermal FSI test case.Lastly, we consider adaptive time-integration for goal-oriented problems, where one is interested in a quantity of interest (QoI), which is a functional of the solution. The state-of-the-art method is the dual-weighted residual (DWR) method, which is extremely costly in both computation and implementation. We develop a goal oriented adaptive method based on local error estimates, which is considerably cheaper in computation. We prove convergence of the error in the QoI for tolerance to zero under a controllability assumption. By analyzing global error propagation with respect to the QoI, we can identify possible issues and make performance predictions. Numerical results verify these results and show our method to be more efficient than the DWR method

Dynamic Analysis of Aerostatic Guideway and FEA Model Development

A dynamically optimal design is essential for a motion system to perform high speed operation without compromising its accuracy, settling time and vibration specification. Good design practice which accounts for dynamic characteristics in the modelling of a motion system warrants higher performance precision machines and cuts down redevelopment effort to ‘patch’ inherent shortcoming of the machine dynamics. This research aimed to accurately describe the non-linear dynamics of a non-mechanical contact aerostatic guideway system in order to achieve an accurate FEA model of the design stage. The single axis aerostatic guideway is comprised of several machine in¬terfaces that impact the dynamic behaviour of the guideway. Modelling each air bear¬ing pad by a single stiffness element is not adequate to predict the guideway modal behaviour accurately. The aerostatic guideway has been broken down into several key machine interface elements. In-depth investigation of the air film and the air bearing mounting mechanism was carried out. A dedicated air film test rig was designed and built to acquire insight of the air film dynamic characteristics. It is observed that the mounting mechanism of the air bearing constitutes to a signifi-cant dynamic effect to the entire air bearing setup. Based on the findings of the mount-ing mechanism’s stiffness properties, a method was developed to estimate ‘true’ air gap heights which cannot be easily assessed and measured directly in most aerostatic guideway carriages. The estimation method enables a more rigorous FE model of the aerostatic guideway system. The comprehensive dynamic analysis methodology pro-posed in this research greatly increases the confidence and accuracy of the aerostatic guideway’s FE model

Effect of surface roughness on the efficiency of self-healing polymers

A shape memory polymer (SMP) is a smart material capable of maintaining two distinct shapes depending on its temperature. A SMP is soft at temperatures above its glass transition temperature but hard below it. When copolyester thermoplastic additives are dispersed in a SMP, it becomes a SMP-based particulate composite capable of self-healing at both the molecular level and the structural level. This makes it very desirable for industrial applications. Upon damage to the composite, the surfaces at the damage interface have to come into contact for efficient healing; the shape memory effect, coupled with a confined recovery (healing) process, ensures this. This study examined the effect of the surface roughness at the damage interface on the efficiency of the healing process. Also studied was the effect of the compressive stress at the point of contact during the healing process on the healing efficiency. The particulate composite (CP-PSMP) consisted of polystyrene shape memory polymer (PSMP) as the matrix and copolyester thermoplastic additives (CP) as the reinforcement. Compressive programming at 10 % pre-strain was performed on the CP-PSMP, which was then tested for its pre-flexural strength. Next, the surfaces were varied using sandpapers of different embedded particle diameters, and the CP-PSMP was healed at 10 MPa using the close-then-heal (CTH) self-healing mechanism. The recovered flexural strength was then obtained and the healing efficiency computed as a fraction of the recovered flexural strength to the pre-healing flexural strength. Healing efficiencies were found to be higher for CP-PSMP with smoother surfaces. The highest healing efficiency of 39 % was found in CP-PSMP with average and root-mean-squared roughness profile parameters, Ra and Rq, of 0.425 and 0.617 μm respectively. Another set of tests revealed that healing was more efficient at higher compressive stresses. Efficiencies at higher compressive stresses (20 – 80 MPa) ranged from 78 % to 118 %. Next, the effects of sanding on healing efficiency was examined by comparing the healing efficiencies of two sets of CP-PSMP with similar Ra and Rq values—one of which was treated with sandpaper. The sanded CP-PSMP samples were 24 % more efficient in healing