Frank Cameron

Variable step size time integration methods for transient eddy current problems

For transient eddy current problems modelled as differential-algebraic equations (DAEs), a time integration method suitable for ordinary differential equations (ODEs) will not necessarily work. We present two Runge-Kutta methods that are suitable for the time integration of the classes of DAEs to which eddy current problems belong. Both methods have error estimators and hence allow variable step sizes. In tests our variable step size integrators were competitive with fixed step size integrators, in particular with Crank-Nicolson.

Quasi stage order conditions for SDIRK methods

The stage order condition is a simplifying assumption that reduces the number of order conditions to be fulfilled when designing a Runge-Kutta (RK) method. Because a DIRK (diagonally implicit RK) method cannot have stage order greater than 1, we introduce quasi stage order conditions and derive some of their properties for DIRKs. We use these conditions to derive a low-order DIRK method with embedded error estimator. Numerical tests with stiff ODEs and DAEs of index 1 and 2 indicate that the method is competitive with other RK methods for low accuracy tolerances.

A class of low order DIRK methods for a class of DAEs

Abstract We study the numerical solution of a DAE described by an implicit differential equation where the state derivative is multiplied by a singular matrix that depends on the state. We consider a class of s-stage DIRK methods having s −1 implicit stages, an explicit first stage and the stiff accuracy property. The DIRKs we consider have global order of at most 3. We determine how many stages are required to meet different order and stability specifications, both for solitary (fixed step size) DIRKs as well as embedded pairs of DIRKs. We present some solitary DIRKs and some embedded DIRK pairs that have appeared in the literature and that are suitable for solving the DAE in question. In addition, we derive some new solitary DIRKs and DIRK pairs. Our tests with embedded pairs show that some pairs may suffer from performance deterioration when the dynamics in the DAE are of different orders of magnitude.

Compressed tracheal sound analysis in screening of sleep-disordered breathing

OBJECTIVE To evaluate the suitability of compressed tracheal sound signal for screening sleep-disordered breathing. METHODS Thirty-three consecutive patients underwent a polysomnography with a tracheal sound analysis. Nineteen patients were healthy except for the sleep complaint, 9 were hypertonic and 3 were hypertonic and had elevated cholesterol. Minimum and maximum values of each consecutive, non-overlapping segment of 15s of original sound data were extracted. All these compressed tracheal sound traces were divided into plain, thin and thick signal periods. Also pure, 10-min episodes of plain, thin and thick tracheal sound periods were selected and the nasal pressure flow shapes during these pure sound periods were examined. RESULTS There was a significant positive correlation between the total nocturnal amount of thick periods and AHI. Apneas and hypopneas were most common during the 10-min episodes of thick sound periods. The proportion of round (normal, non-flattened) inspiratory flow shape was highest during the pure plain periods. CONCLUSIONS Breathing consisting of apneas and hypopneas can quite reliably be visualised with compressed tracheal sound analysis. The other interesting outcome of the study is that even prolonged flow limitation might be revealed with the method. SIGNIFICANCE Compressed tracheal sound analysis might provide a promising screening method for obstructive apneas and hypopneas.

Automatic generation of efficient routines for evaluating multivariate polynomials arising in finite element computations

Abstract Multivariate polynomials commonly occur in finite element computations. Horners method is an efficient way of evaluating multivariate polynomials and their derivatives. We have developed MAPLE procedures that generate efficient C or FORTRAN routines based on Horners method. Operation counts for some common finite element computations are shown to be significantly lower with these routines than with some other codes from the literature.

A Matlab package for automatically generating Runge-Kutta trees, order conditions, and truncation error coefficients

In designing parts of Runge-Kutta methods, order conditions and truncation error coefficients (TECs) are needed. Order conditions and TECs are typically presented as a set of trees combined with rules for producing algebraic expressions from the trees. The tree sets are defined recursively and can be generated by hand only for low orders. This article describes a package of Matlab routines for automatically generating Runge-Kutta trees, order conditions, and TECs. The routines are capable of generating Maple code, Matlab code, or LaTeX expressions for ODEs or DAEs of index 1 and 2. In producing the package, two theoretical problems are tackled: (a) avoiding the repeated generation of the same tree and (b) the efficient storage of TECs.

Towards Systematic Compliance Evaluation Using Safety-Oriented Process Lines and Evidence Mapping

The role of software is growing in safety related systems. This underlines the need for software process assessment in many safety-critical domains. For example, the nuclear power industry has strict safety requirements for control systems and many methods are applied to evaluate compliance to domain specific standards and requirements. This paper discusses the needs of the nuclear domain and presents alternatives to develop a process assessment method that takes into account domain specific requirements. The aim is to provide an approach that facilitates the use of assessment findings in evaluating compliance with the domain requirements and supports other assurance needs. Safety-oriented Process Line Engineering (SoPLE) is studied as a method for mapping assessment criteria to domain specific requirements. A binary distance metric is used to evaluate, how far a process mapping based method would solve problems found in compliance evaluation. Based on the results, SoPLE is applicable in this case, but process mapping is not adequate to facilitate compliance evaluation.

Heuristic Methods for Assigning Spent Nuclear Fuel Assemblies to Canisters for Final Disposal

Abstract The disposal of spent fuel assemblies (SFAs) by companies currently producing nuclear power in Finland is the responsibility of a company named Posiva Oy. Posiva Oy has decided to use the KBS-3 (Swedish abbreviation for nuclear fuel safety; version 3) concept. In KBS-3, SFAs are placed in metal canisters, which are themselves deposited deep into crystalline rock. The disposal process in Finland will last many decades. To efficiently assign SFAs to canisters, in this paper we study the minimax canister formation problem. In this problem, we assume we are given two sets of data: (a) a schedule specifying the number of disposal canisters per year and (b) the decay heat of each SFA for every disposal year. The goal in the problem is to assign SFAs to canisters so that the largest canister heat load is minimized. The minimax canister formation problem is a variant of a well-known optimization problem: makespan minimization on unrelated parallel machines. We developed heuristic methods for solving the minimax canister formation problem. Using our methods and predicted SFA amounts and properties for Finland, we obtained high-quality solutions in numerous test cases. We also investigated how the uncertainty in SFA burnups affects the canister heat loads.

Validating Effectiveness of Compliance Evaluation in Safety-Critical Process Assessment

Safety is a growing concern in systems containing software. Process assessments are used to ensure quality of the development processes for safety-critical systems. Regulators of the nuclear power industry have increased requirements for the development processes. Now, process assessments are integrated in the qualification of the systems and are expected to provide information not only based on the assessment model but also about the domain specific requirements. Compliance evaluation utilizes assessment findings to create evidence of the relevant domain specific requirements. This study aims at validating the results of compliance evaluation by analyzing assessment findings and their use in the context of a safety-critical domain. The analysis is based on the use of binary relations. The results of this analysis indicate that process assessment findings are also relevant when evaluating compliance to other sets of requirements. The results can significantly support the trustworthiness of the process assessment method and ensure effective use of process assessments.

Two space-saving algorithms for computing the permuted transpose of a sparse matrix

Abstract Given an m×n sparse matrix A having na nonzeros and a permutation matrix P, we consider the problem of finding (PA)T in a space-saving manner. Two algorithms, TRANSPERM1 and TRANSPERM2, are presented. Both algorithms make use of the in-place inversion of permutation vectors. To save even more space TRANSPERM1 forms a piecewise linear model of the nonzero distribution. The computational complexity of TRANSPERM2 is O(na,n,m). For a particularly treachrous nonzero distribution we derive an upper bound for the complexity of TRANSPERM1 of O(na,τnan/nl,m,n,nl), where nl is the number of intervals in the piecewise model and τ≤ 1 2 . These two algorithms were compared with Gustavsons efficient HALFPERM algorithm.3 We expected our algorithms to be slower than HALFPERM since they were written with the intention of saving space. In tests on 21 matrices, HALFPERM was on average 2·5 times faster than TRANSPERM1 and 1·6 times faster than TRANSPERM2. However, for these 21 matrices, TRANSPERM1s storage requirements were on average only 51% of HALFPERMs, and TRANSPERM2s requirements were only 67% of HALFPERMs.