In this thesis, we present a method for the automated generation of numerical evaluation routines for bivariate functions via tensor product series and develop a toolkit to assist with the generation of the approximations. The final approximations can be evaluated in user-defined precision or in hardware floating point precision by default. The evaluation routines can also be compiled into a C library (or a library in some other language) for more efficient evaluations. The toolkit can be used for various mathematical functions of two variables, such as Bessel functions or user-defined functions, at any given precision. The method of tensor product series expansion reduces the bivariate approximation problem to a sequence of univariate approximation problems. In order to control the degrees of the approximating functions so that evaluation will be accurate and efficient, we recursively divide the bivariate intervals into subintervals until both the number of terms in the tensor product series and the degrees of the univariate approximations are less than specified bounds. We then generate in each subinterval rational approximations using Chebyshev-Padé approximants or polynomial approximations using Chebyshev series according to the user's specification. Finally we show the experimental results for a variety of bivariate functions, which achieve a significant speedup over the original Maple functions for evaluation in hardware floating point precision. We also compare the results of choosing polynomial approximations versus rational approximations for the univariate subproblems.

In this thesis, we present a method for the automated generation of numerical evaluation routines for bivariate functions via tensor product series and develop a toolkit to assist with the generation of the approximations. The final approximations can be evaluated in user-defined precision or in hardware floating point precision by default. The evaluation routines can also be compiled into a C library (or a library in some other language) for more efficient evaluations. The toolkit can be used for various mathematical functions of two variables, such as Bessel functions or user-defined functions, at any given precision. The method of tensor product series expansion reduces the bivariate approximation problem to a sequence of univariate approximation problems. In order to control the degrees of the approximating functions so that evaluation will be accurate and efficient, we recursively divide the bivariate intervals into subintervals until both the number of terms in the tensor product series and the degrees of the univariate approximations are less than specified bounds. We then generate in each subinterval rational approximations using Chebyshev-Padé approximants or polynomial approximations using Chebyshev series according to the user's specification. Finally we show the experimental results for a variety of bivariate functions, which achieve a significant speedup over the original Maple functions for evaluation in hardware floating point precision. We also compare the results of choosing polynomial approximations versus rational approximations for the univariate subproblems.

Thesis (MScEng (Civil Engineering))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: End stop impact forces are horizontal longitudinal forces imposed by the crane on the end stops. Both the previous South African loading code SABS 0160:1989 and the current South African loading code SANS 10160 , classify end stop impact force as an accidental load case , hence they are not expected to occur within the expected lifetime when the guide lines for crane operation are strictly adhered to. In the estimation of end stop impact force, the previous South African loading code SABS 0160:1989 gives two guidelines for estimating the end stop impact force. The first guideline is simplistic in its approach and it’s based on the assumption that the crane and its supporting structure act as rigid bodies; hence calculation is based on rigid body mechanics. Literature reviewed reveals that this is not correct. The second guideline is more explicit in its approach as it takes into account the crane speed, resilience of the buffers and resilience of the end stops. The current South African loading code, SANS 10160 gives a better representation of the dynamics of the crane movement. However, the dynamic factor recommended for the estimation of end stop impact force is empirical in nature and thus lacks adequate scientific backing. One of the purposes of this research was to investigate the influence of the stiffness of the crane bridge on the end stop impact force. This was achieved by conducting a series of FEA simulations on the double bridge EOHTC fitted with elastomeric buffers. For this set of simulations, the effect of each influencing parameter on the end stop impact force was investigated, and the maximum end stop impact force was obtained using a constraint optimization technique. From the results obtained, comparison was then made with the existing maximum end stop impact force for a single bridge EOHTC fitted with elastomeric buffers. Another purpose of this research was to investigate the end stop impact force for an electric overhead travelling cranes (EOHTC) fitted with hydraulic buffers taking into account the dynamics involved in the movement of the EOHTC. This was achieved by a series of experimental and numerical investigation. The numerical investigation was conducted using an existing numerical model of an EOHTC which captures the crane and its supporting structure as a coupled system. Finite element analysis (FEA) impact force histories obtained were calibrated to the base experimental impact force histories. Thereafter, a series of FEA simulations were conducted by changing the parameters which have a substantial effect on the end stop impact forces. This yielded various maximum impact peaks for various parameters. The maximum impact force was then mathematical obtained from the FEA impact force histories for a given level of reliability using a constraint optimization technique. Also, codified end stop impact forces were calculated for the SABS 0160:1989 and SANS 10160-6:2010. From the results obtained, comparison was made between the codified end stop impact force and the maximum impact force obtained from the constraint optimization technique. / AFRIKAANSE OPSOMMING: Ent buffer impak kragte is horisontale kragte wat deur die kraan op die entbuffers aangewend word. Beide die Suid Afrikaanse las kode SABS 0160:1989 en die voorgestelde Suid Afrikaanse las kode SANS 10160, klasifisseer die entbuffer impak kragte as ‘n ongeluks las geval, dus word die kragte nie verwag tydens die verwagte leeftyd van die kraan wanneer die riglyne van die kraan prosedures streng gevolg word nie. Volgens die Suid-Afrikaanse las kode SABS 0160:1989 word daar twee riglyne voorgestel om die entbuffer kragte te bepaal. Die eerste riglyn is ‘n eenvoudige riglyn en word gebaseer op die aaname dat die kraan en die ondersteunende struktuur as ‘n starre ligame reageer en dus word die kragte bereken deur star ligaam meganika, alhoewel, uit die literatuur word dit bewys as inkorrek. Die tweede riglyn is ‘n meer implisiete benadering aangesien dit die kraan snelheid, elastisiteit van die buffers sowel as die elastisiteit van die end stoppe in ag neem. SANS 10160-6:2019 gee ‘n beter benadering van die dinamiese beweging van die kraan. Die voorgestelde dinamiese faktor waarmee die ent_buffer_kragte bereken word, is empiries van natuur . Een van die doelstellings vir die navorsings projek was om te bepaal wat die invloed van die kraan brug se styfheid op die entbuffer kragte is. ‘n Aantal Eindige Element Analise (FEA) simulasies is uitgevoer op ‘n dubbel brug elektriese aangedrewe oorhoofse kraan met elastomeriese buffers. Van die stel FEA simulasies kan die invloed van elke parameter op die entbuffer impak_kragte bepaal word. Die maksimum entbuffer impak_kragte is bepaal met behulp van ‘n beperking optimiserings tegniek. Vanaf hierdie resultate is ‘n vergelyking gemaak met die bestaande maksimum ent_buffer impak_kragte vir ‘n enkel brug elektriese oorhoofse aangdrewe kraan met elastomeriese buffers. ‘n Tweede doel rede vir die navorsing was om te bepaal wat die ent buffer impak_kragte op ‘n elektriese aangedrewe oorhoofse kraan met hidrouliese buffers is. Dit is bepaal deur ‘n aantal eksperimentele en numeriese toetse uit te voer. Die numeriese toetse is uitgevoer deur gebruik te maak van ‘n huidige numeriese model van ‘n elektriese aangedrewe oorhoofse kraan wat die kraan en die ondersteunende struktuur as ‘n. Die Eindige Element Analise impak_kragte is gekalibreer teen die eksperimenteel bepaalde impak- kragte. Daarna is ‘n reeks Eindige Element Analise simulasies uitgevoer en sodoende die parameters te verander wat die mees beduidende invloed op die end stop impak_kragte het. Dit het verskeie impak_krag pieke vir verskillende parameters meegebring. Die maksimum impak kragte is bepaal van die impak kragte van die Eindige Element Analise vir ‘n gegewe vlak van betroubaarheid deur gebruik te maak van die beperking optimiserings tegniek. Daarmee saam is die gekodifiseerde ent buffer impak kragte bereken volgen SABS 0160:1989 en die SANS 10160- 6:2010. Vanaf hierdie resultate is ‘n vergelyking gemaak tussen die gekodifiseerde entbuffer impak_kragte en die maksimum impak_kragte wat bepaal is deur die (beperking optimiserings tegniek).

<p>Ova doktorska teza analizira problem re&scaron;avanja karakterističnih<br />jednačina, koje se koriste prilikom re&scaron;avanja jednačina svojstvenih<br />oblika, definisanih kroz harmonijski spojen metod konačnih<br />traka. U slučaju složenijih graničnih uslova pokazano<br />je da gre&scaron;ke određivanja korenova karakteristične jednačine rastu<br />gotovo eksponencijalno sa svakim narednim modom, usled<br />prirode s&acirc;mih karakterističnih jednačina. Tako dobijeni korenovi<br />karakteristične jednačine značajno i nepovoljno utiču na<br />tačnost re&scaron;avanja jednačina svojstvenih oblika, izračunavanja<br />određenih integrala, kao i celokupnih proračuna. Predstavljena<br />je hibridna metoda putem koje se na pouzdan nacin određuju<br />korenovi karakterističnih jednačina, re&scaron;avaju jednačine svojstvenih<br />oblika i izračunavaju određeni integrali. Razvijen je<br />prototip referentne Open Source implementacije hibridne metode,<br />uz podsistem za automatsku verifikaciju koji rigorozno<br />verifikuje karakteristike hibridne metode i njene referentne implementacije,<br />za sve navedene granične uslove i integrale.</p> / <p>This PhD thesis analyzes the problem of solving the characteristic<br />equations of the basic functions, as defined by the harmonic<br />coupled finite strip method. It&rsquo;s found that with each increasing<br />mode the characteristic equation root-finding error grows<br />exponentially for all but the most trivial edge boundary conditions,<br />due to the hyperbolic functions involved. These large rootfinding<br />errors will lead to severe accuracy issues when computing<br />basic functions and their integrals, especially for higher modes.<br />A hybrid method for accurately solving characteristic equations<br />and obtaining the required integrals is presented, along with its<br />reference Open Source implementation. An extensive test suite<br />has been developed to verify the hybrid method and its implementation<br />for all the presented boundary conditions and integrals.</p>