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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Calcified tissue international 10 (1972), S. 82-90 
    ISSN: 1432-0827
    Keywords: Calcium ; Phosphate ; Precipitation ; Kinetics ; Chemistry
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Medicine , Physics
    Description / Table of Contents: Résumé La cinétique de la formation et de la transformation des précipités de phosphate de calcium, obtenus en mélangeant de volumes égaux de solutions à 6×10−3 M de calcium total et/ou phosphate total est étudiée à 25°C. Les solutions de phosphate sont préajustées à un pH de 7.4. Les changements de pH et de turbidité des solutions sont suivis simultanément en fonction du temps. Les précipités sont isolés à des intervalles de temps variables et caractérisés par diverses méthodes physico-chimiques. Initialement un précipité avec un rapport molaire Ca/P de 1.5, amorphe aux rayons X et en diffraction électronique, est formé. Le spectre IR indique la présence de PO 4 3− et de HPO 4 2− . Après une période métastable, on observe la précipitation d'un matériel cristallin dans ou sur la phase amorphe. Vingt quatre heures après préparation de l'échantillon les précipités présentent surtout les caractères du phosphate octocalcique.
    Abstract: Zusammenfassung Die Kinetik der Bildung und Transformation von Calciumphosphat-Niederschlägen wurde bei 25°C untersucht. Es wurden dazu gleiche Volumen von Lösungen gemischt, bei einer Konzentration von 6×10−3M totales Calcium und/oder totales Phosphat. Die Phosphatlösungen wurden zuerst auf pH 7,4 eingestellt. Veränderungen des pH und Trübung der Lösungen wurden gleichzeitig als eine Funktion der Zeit aufgezeichnet. Niederschläge wurden in verschiedenen Zeitintervallen isoliert und mit verschiedenen physiko-chemischen Methoden charakterisiert. Am Anfang wurde ein Niederschlag mit einem molaren Ca/P-Verhältnis von 1,5, im Röntgenbild und in der Elektronendiffraktion amorph, gebildet. Infrarotspektren deuteten die Anwesenheit von PO 4 3− - und HPO 4 2− -Ionen an. Nach einer metastabilen Periode erfolgte ein Niederschlag aus kristallinem Material innerhalb oder auf der amorphen Substanz. 24 Std nach der Herstellung der Proben zeigten die Niederschläge in der Hauptsache die Charakteristiken von Octocalciumphosphat.
    Notes: Abstract The kinetics of the formation and transformation of calcium phosphate precipitates obtained by mixing equal volumes of solutions, 6×10−3 M in total calcium and/or total phosphate was investigated at 25°. The phosphate solutions were preadjusted to pH 7.4. Changes of the pH and turbidity of the solutions were followed simultaneously as a function of time. Precipitates were isolated at various time intervals and characterized by different physicochemical methods. Initially a precipitate with a molar Ca/P ratio of 1.5, amorphous to X-ray and electron diffraction was formed. IR spectra indicated the presence of PO 4 3− and HPO 4 2− ions. After a period of metastability, precipitation of a crystalline material within or upon the amorphous matter occurred. Twenty four hours after sample preparation the precipitates showed mainly the characteristics of octacalcium phosphate.
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  • 2
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    Springer
    Calcified tissue international 10 (1972), S. 38-48 
    ISSN: 1432-0827
    Keywords: Bivalve ; Molluse ; Shell ; Carbonates ; Ultrastructure
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Medicine , Physics
    Description / Table of Contents: Résumé La technique du bombardement à l'aide d'ion d'argon est utilisée pour réduire l'épaisseur de la coquille de carbonate de calcium des bivalvesMytilus etMercenaria pour examen au microscope électronique par transmission et en diffraction électronique; une comparaison est réalisée à l'aide de répliques simples, servant de témoins. Les résultats obtenus confirment les études antérieures de répliques et de microscopie par balayage. De plus, une structure “aérée” est mise en évidence dans la coquille des aragonites, et surtout dans le nacre deMytilus. Cette structure est interprêtée comme un artefact induit par la chaleur, formé par l'inclusion d'eau et de matériel organique, interprétation qui concorde avec les études chimiques et de microscopie électronique.
    Abstract: Zusammenfassung Die Beschießung mit Argonionen wurde angewendet, um die Dicke von Calciumcarbonat-Schalen der zweischaligen MuschelnMytilus undMercenaria zu reduzieren. Diese Technik erlaubte die Ausführung von Transmissions-Elektronenmikroskopie und Elektronendiffraktion, wobei gleiche Proben nach einer bereits bestehenden Methode vorbereitet und als Kontrollen herangezogen wurden. Es wurden zusätzliche Resultate zu den Muschelstruktur-studien erhalten, welche früher publizierte Arbeiten unterstützen, die mit der Abklatschmethode und der Raster-Elektronenmikroskopie ausgeführt worden waren. Zusätzlich wurde eine „schaumartige” Struktur der Muschelaragoniten, besonders im Perlmutter vonMytilus, beobachtet. Da es sich um ein durch Hitze verursachtes Artefakt handelt, wird diese Struktur als Einschlüsse von Wasser und organischem Material interpretiert, was den Befunden von verschiedenen veröffentlichten chemischen und elektronenmikroskopischen Arbeiten entspricht.
    Notes: Abstract Use is made of the argon ion-bombardment technique to reduce the thickness of calcium carbonate shells of the bivalvesMytilus andMercenaria for transmission electron microscopy and electron diffraction, with comparison of single-stage replicas of similar specimens serving as controls. As an additional approach to shell structure studies, it gives results which support earlier published work with both replicas and scanning microscopy. In addition, a “frothy” structure is detected in the shell aragonites, especially inMytilus nacre. As a heat-induced artifact, it is interpreted as representing trapped water and organic material inclusions, an interpretation consistent with several published chemical and electron microscope studies.
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  • 3
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    Calcified tissue international 10 (1972), S. 216-222 
    ISSN: 1432-0827
    Keywords: Ultrastructure ; Callus ; Osteoclast ; Endocytosis ; Lysosomes
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Medicine , Physics
    Description / Table of Contents: Résumé La bordure en brosse des ostéoclastes de cals de fractures de rats présente des plissements complexes de la membrane cytoplasmique formant des canaux étroits. L'absorption d'un produit exogène opaque aux électrons (des macromolécules de dioxyde de thorium) s'effectue par l'intermédiaire de ces canaux, par un «courant» membranaire. Les contenus des canaux sont transférés à des lysosomes («granules spécifiques»), situés sous la bordure en brosse. Dans des «régions de transition», adjacentes à cette dernière, l'absorption de dioxyde de thorium se fait par «vésiculation membranaire» (endocytose classique).
    Abstract: Zusammenfassung Der gekrauste Rand der Osteoklasten im Frakturcallus von Ratten besteht aus komplexen Einstülpungen der Plasmamembran, die enge Kanälchen bildet. Die Absorption einer exogenen, elektronisch dichten Verbindung, Thoriumdioxyd, erfolgt durch diese Kanäle, offenbar durch einen „Membranfluß”. Der Inhalt der Kanäle wird zu den Lysosomen („spezifische Granula”) geführt, welche unter dem gekrausten Rand liegen. In „Übergangsgebieten”, welche sich neben dem gekrausten Rand befinden, scheint die Aufnahme der Thoriumdioxydpartikel durch „Bläschenbildung in der Membran” (konventionelle Endocytose) stattzufinden.
    Notes: Abstract The ruffled border of osteoclasts in the fracture callus of rat consists of complex infoldings of the plasma membrane forming narrow channels. Absorption of an exogenous, electron-dense compound, thorium dioxide, has been shown to take place via these channels, apparently through “membrane flow”. The contents of the channels are transferred to lysosomes (“specific granules”) located subjacent to the ruffled border. In “transitional regions” adjacent to the ruffled border, uptake of thorium dioxide particles appeared to occur through “membrane vesiculation” (conventional endocytosis).
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  • 4
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    Calcified tissue international 9 (1972), S. 296-309 
    ISSN: 1432-0827
    Keywords: Osteoclasts ; Enzyme ; Parathyroid ; Microdissection ; Ultrastructure
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Medicine , Physics
    Description / Table of Contents: Résumé Une nouvelle méthode d'isolement d'ostéoclastes est mise au point pour des analyses biochimiques et de microscopie électronique. Pour isoler les cellules par microdissection, des empreintes d'os métaphysaire sont utilisées. Cette méthode, supérieure aux coupes d'os, permet une meilleure préservation cytologique et enzymatique et permet d'obtenir des cellules totales plus faciles à manipuler, avec des résultats plus reproductibles. Par analyse planimètrique de cellules isolées, colorées histochimiquement, il apparait que les ostéoclastes constituent plus de 90% de la masse de l'échantillon. Les concentrations de la phosphatase acide et de certaines enzymes, liées au nucléotide pyridinique, entrant dans le métabolisme de l'acide citrique, sont déterminées dans des échantillons d'ostéoclastes, pesant de 0,2 à 2,0 μg, isolés à partir de rats normaux et parathyroidectomisés. L'activité en aconitase, mesurée en direction de la transformation de citrate en isocitrate, est de 0,5–0,8 M/Kd/H, la plus faible des activités étudiées. Les activités en GDH et NADP-ICDH sont 5 à 10 fois supérieures que celle de l'aconitase, mais seulement un dixième à un tiers de celle de la phosphatase acide, de la déshydrogénase lactique ou malique.
    Abstract: Zusammenfassung Es wird eine neue Technik beschrieben, welche die Isolierung von Osteoklasten für biochemische und elektronenmikroskopische Untersuchungen ermöglicht. Als Ausgangsmaterial zur Zellisolierung wurden Abstriche von Metaphysenknochen benützt. Die Verwendung von Abstrichen bietet gegenüber Knochenschnitten wichtige Vorteile, wie z.B. eine bessere Erhaltung der cytologischen und enzymatischen Eigenschaften sowie die Gewinnung von unverletzten Zellen, welche leichter verarbeitet werden können und besser reproduzierbare Daten ergeben. Durch planimetrische Analyse der histochemisch gefärbten Ausstriche von isolierten Zellen konnte nachgewiesen werden, daß die Osteoklasten über 90% des gesamten Probenmaterials ausmachen. Die Mengen verschiedener Enzyme, welche an Pyridinnukleotid gebunden und am Citronensäuremetabolismus beteiligt sind, sowie der sauren Phosphatase wurden in Osteoklastenproben bestimmt, welche ein Gewicht von 0,2–2,0 μg hatten und aus Knochen von normalen und mit Parathyroidextrakten behandelten Ratten isoliert worden waren. Die Aktivität der Aconitase, welche in der Richtung von Citrat zu Isocitrat gemessen wurde, war mit 0,5–0,8 M/Kd/H die niedrigste aller untersuchten Aktivitäten. Die Aktivitäten der GDH und der NADP-ICDH waren 5–10mal höher als jene der Aconitase, entsprachen jedoch nur einem Zehntel bis einem Drittel derjenigen der sauren Phosphatase, der Laktat- oder der Malatdehydrogenase.
    Notes: Abstract A new method is described by which osteoclasts can be isolated for biochemical and electron microscopic analyses. As a source of cells for isolation by microdissection, imprints of metaphyseal bone were used. The use of imprints provides important advantages over bone sections, including a higher degree of cytologic and enzymatic preservation, and the delivery of whole cells which are more readily manipulated and which yield data that are more readily reproduced. By planimetric analysis of the histochemically-stained isolated cell samples, it was shown that osteoclasts represent over 90% of the sample mass. The levels of several of the pyridine nucleotide-linked enzymes involved in citric acid metabolism, as well as acid phosphatase, were determined in osteoclast samples weighing 0.2 to 2.0 μg isolated from normal and parathyroid-treated rats. Aconitase activity measured in the direction of citrate to isocitrate was 0.5–0.8 M/Kd/H, the lowest of the activities studied. The activities of GDH and NADP-ICDH were 5 to 10 times higher than that of aconitase but only a tenth to a third that of acid phosphatase, lactic or malic dehydrogenase.
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  • 5
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    Calcified tissue international 10 (1972), S. 171-197 
    ISSN: 1432-0827
    Keywords: Amorphous ; Crystalline ; Calcium phosphate ; Chemistry ; Composition
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Medicine , Physics
    Description / Table of Contents: Résumé Des échantillons non lavés de phosphate de calcium amorphe (ACP) contiennent une fraction labile, non remplaçable, riche en phosphate acide avec un rapport Ca/P faible: cette fraction est perdue de façon irréversible au cours du lavage. De l'ACP frais, précipité entre pH 6.6–10.6, varie dans un rapport molaire Ca/P de 1.18 à 1.50 et dans un rapport HPO 4 2− /P total de 33.0% à 10.1%. A pH 7.40, de l'ACP frais a un rapport molaire Ca/P de 1.36±0.02 et contient 22.8 (±2.2)% HPO 4 2− . Les résultats obtenus avec du précipité non lavé ne peuvent s'expliquer par du Ca2+ emprisonné et de l'HPO 4 2− ou du Na+, Cl− et CO 3 2− exogènes. Les phosphates de calcium amorphes constituent une classe de sels ayant des caractères chimiques variables et des propriétés physiques identiques, comparables au verre. Le CaHPO4·xH2O non cristallin peut être un ACP, surtout au cours des phases précoces de formation. A des pH physiologiques, l'ACP se transforme en petits cristaux applatis contenant de fortes quantités de phosphate acide facilement remplaçable. Le fait de laver la couche de surface produit un changement chimique dans les nouveaux cristaux: des cristaux non lavés donnent des diagrammes de diffraction d'apatite peu cristallins, ainsi que des spectres infra-rouges peu nets, intermédiaires entre des apatites et du phosphate octocalcique. Des explications structurales sont proposées et les compositions minérales amorphe/cristalline de l'os et du cartilage sont recalculées.
    Abstract: Zusammenfassung Ungewaschene Proben von amorphem Calciumphosphat (ACP) enthalten eine unersetzliche labile Fraktion, welche reich an saurem Phosphat ist und ein niederes Ca/P-Verhältnis hat und welche während des Waschprozesses unwiderruflich verloren geht. Natives ACP, welches im pH-Bereich 6,6–10,6 ausgefällt wurde, variierte im molaren Ca/P-Verhältnis zwischen 1,18 und 1,50 und in HPO 4 2− /totales P zwischen 33,0 und 10,1%. Bei pH 7,40 hatte natives ACP ein molares Ca/P-Verhältnis von 1,36±0,02 und enthielt 22,8 (±2,2)% HPO 4 2− . Die Werte beim ungewaschenen Niederschlag rühren weder von aus dem Überstand aufgenommenem Ca2+ und HPO2−, noch von außen kommendem Na+, Cl− und CO 3 2− her. Die amorphen Calciumphosphate werden als eine Klasse von Salzen erkannt, welche veränderliche chemische, aber identische glasartige physicochemische Eigenschaften haben. Nicht kristallines CaHPO4·xH2O kann auch ein ACP sein, besonders in den frühen Bildungsstadien. Bei physiologischem pH verwandelt sich ACP in kleine plattenförmige Kristalle, welche große Mengen von leicht ersetzbarem saurem Phosphat enthalten. Das Waschen dieser Oberflächenschicht erzeugte chemische Veränderungen in den resultierenden Kristallen; ungewaschene Kristalle zeigten ein Diffraktionsmuster, das nur schwach demjenigen des kristallinen Aspatites glich, aber ein schlecht aufgelöstes Infrarotspektrum, welches zwischen Apatit und Octocalciumphosphat war. Es werden strukturelle Erklärungen für alle diese Phenomena diskutiert, und revidierte amorph/kristalline Mineralzusammensetzungen von Knochen und Knorpel wurden neu berechnet.
    Notes: Abstract Unwashed samples of amorphous calcium phosphate (ACP) contain an irreplaceable labile fraction, rich in acid phosphate and low in Ca/P ratio, which is irreversibly lost during the washing process. Native ACP precipitated in the pH range 6.6–10.6 varied in Ca/P molar ratio from 1.18 to 1.50 and in HPO 4 2− /total P from 33.0% to 10.1%. At pH 7.40, native ACP had a Ca/P molar ratio of 1.36±0.02 and contained 22.8 (±2.2)% HPO 4 2− . Unwashed precipitate data could not be attributed to either trapped supernatant Ca2+ and HPO 4 2− or extraneous Na+, Cl−, and CO 3 2− . The amorphous calcium phosphates are recognized as a class of salts having variable chemical but identical glass-like, physicochemical properties. Non-crystalline CaHPO4·xH2O may also be an ACP, especially during early formative stages. At physiological pH, ACP transforms to small platy crystals containing large amounts of readily-replaceable acid phosphate. Washing this surface layer produced chemical alterations in the resultant crystals; unwashed crystals had poorly-crystalline apatitic diffraction patterns but exhibited poorly-resolved infrared spectra intermediate between apatite and octacalcium phosphate. Structural explanations for all these phenomena are discussed, and revised bone and cartilage amorphous/crystalline mineral compositions have been re-calculated.
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  • 6
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    Calcified tissue international 9 (1972), S. 238-242 
    ISSN: 1432-0827
    Keywords: Dentine ; Ultrastructure ; Tubule ; Tooth
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Medicine , Physics
    Description / Table of Contents: Résumé L'étude ultrastructurale de la dentine humaine périphérique, de couronnes dentaires de sujets âgés de 11 à 75 ans, a montré trois aspects principaux du contenu des canalicules en coupe transversale. Il s'agit de canalicules apparemment vides, de canalicules contenant un matériel organique annulaire et, enfin, de canalicules totalement remplis d'un matériel organique d'aspect granulaire ou hyalin. Aucune terminaison nerveuse n'est visible à ce niveau.
    Abstract: Zusammenfassung Die ultrastrukturelle Untersuchung von peripherem menschlichem Zahnkronendentin bei Patienten im Alter von 11–75 Jahren hat drei Hauptaspekte des Inhaltes der Dentintubuli gezeigt. Sie bestehen bei transversalen Schnitten aus toten Gängen sowie beim Lumen der Tubuli entweder aus ringförmigen oder ganzausfüllenden Ablagerungen. Im äußeren Dentin wurden keine Nervenendigungen beobachtet.
    Notes: Abstract An ultrastructural study of peripheral human coronal dentin in patients aged 11 to 75 years, has shown main aspects of the dentinal tubular content. In transverse sections, they consist of dead tracts and annular or solid content to the tubular lumen. No nerve endings were observed in the outer dentin.
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  • 7
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    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 4 (1972) 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
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  • 8
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 1-3 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
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  • 9
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 5-10 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Two methods for the numerical evaluation of orthogonal damping matrices are developed. The first relates the modal damping ratios to the coefficients of the Caughey series. The second is a direct approach which expresses the damping matrix as a sum of a series of matrices each of which produces damping in a particular mode. It is found that the direct approach is more convenient to apply and is less numerically sensitive than the series approach.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 67-84 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A mixed triangular finite element model has been developed for plate bending problems in which effects of shear deformation are included. Linear distribution for all variables is assumed and the matrix equation is obtained through Reissner's variational principle. In this model, interelement compatibility is completely satisfied whereas the governing equations within the element are satisfied ‘in the mean’. A detailed error analysis is made and convergence of the scheme is proved. Numerical examples of thin and moderately thick plates are presented.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 133-142 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: This paper presents a sector shape of an element ideally suited for the analysis of sectorial and annular plates. A conforming function with twenty degrees of freedom is developed in polar co-ordinates to represent the deformation of the element in bending. The finite element method is applied to some plate bending problems to illustrate the power and efficiency of sector elements.
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  • 12
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 143-144 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
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  • 13
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 144-145 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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  • 14
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 148-151 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 145-148 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
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  • 16
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 151-151 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
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  • 17
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 195-206 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Five previously employed numerical methods for the solution of Orr-Sommerfeld problems have been compared to each other and to a new method, the differential method of near-orthonormalized integration. Brief summaries of each method are included. The comparison, based on seven factors, reflects the results of an implementation of a computer program for each method for the classic Orr-Sommerfeld problem of plane Poiseuille flow. This comparison shows that the new method and the algebraic finite difference method are currently the best available numerical solution methods for the problems in this class, with the new method being less problem dependent.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 207-216 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In this paper, a method is described which allows a direct derivation of a set of first-order finite difference equations to numerically compute the motion of any conservative or non-conservative dynamic system with a finite number of degrees of freedom. The derivation of the method is based on an application of Lagrangian multipliers to a functional form of Hamilton's equations, and reduces the work required to obtain the most desirable form for numerical integration from the standpoint of computational efficiency and accuracy. For systems with many degrees of freedom, the required matrix inversions produce first derivatives of the co-ordinates, instead of second derivatives, thus eliminating a potential source of error in numerical integration. Two examples are given to illustrate the method.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 235-249 
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    Keywords: Engineering ; Engineering General
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    Notes: This paper presents the derivation of the stiffness matrix for a lumped parameter hyperbolic paraboloid shell element. This element is used to obtain solutions to hyperbolic paraboloid shells with various edge conditions. Numerical example for three types of support conditions (a simple support, a clamped edge and a shell supported by concentric edge beams) are presented. To illustrate the applicability of the method, the results obtained by the lumped parameter element are compared with those obtained by well-known methods of analysis such as the variational procedure, the finite difference and the finite element method.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 217-234 
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    Keywords: Engineering ; Engineering General
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    Notes: The convergence proof of plate eigenvalue solution from conforming displacement finite elements is presented. The analysis is based on converting a thick plate free vibration problem in to a corresponding isoperimetric variational problem. A conforming thick, plate element is used to illustrate the mathematical development. On the basis of the derived asymptotic rate of convergence of the approximate eigenvalues, the authors propose a practical method of improving the numerical solutions. Extension of the mathematical proof to cover classical thin plate finite elements is briefly discussed.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 251-260 
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    Notes: Two-dimensional stagnation point flow of a viscous fluid is modified by the introduction of a thin plate placed symmetrically at the stagnation point. Separation of the boundary layer results in back flow near the plate even at small Reynolds numbers. The Navier-Stokes equations governing the flow are replaced by finite differences using unequal net spacing which is finer in the boundary layer. The choice of finite differences guarantees convergence of the iterative procedures used. Only those relaxation parameters, which are necessary to obtain convergence and also to make the method computationally efficient, have been used. Numerical results for Reynolds numbers 9, 100 and 400 are obtained.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 261-277 
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    Notes: A mixed variational formulation is used as basis for developing a mixed finite element method for axisymmetric shell. The independent unknowns of the method are the axial and radial displacement components, the rotation of the normal to the middle surface and the meridional bending stress couple. The basic element is a frustrum of curved meridian. General advantages of the mixed method are presented, one of which is the possibility of using piece-wise linear functions of the meridional arclength to represent the basic unknowns. Test results are presented for plate bending, transverse shear deformation, membrane behaviour, edge-zone bending, bending near the junction of two shells, convergence of the method and accuracy of middle surface curvature interpolation. Shell geometries considered are cylindrical, conical, spherical and ellipsoidal. Good results are obtained which should increase interest in the relatively less known and less tested mixed method as compared to the stiffness method.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 279-287 
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    Keywords: Engineering ; Engineering General
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    Notes: A generalized finite difference scheme is developed to represent the one-dimensional consolidation process. The difference equations are derived from integral balance equations which represent the soil as a mixture of two constituents. No restrictions are placed on the soil constitutive properties or stratigraphy. An approach to stability analysis for the scheme is outlined, and an example problem is presented.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 289-299 
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    Notes: The problem of an impulsively applied pressure acting on the surface of a spherical cavity in a linear viscoelastic medium is solved by an approximate inversion of the Fourier transform. The method can be applied to general models of viscoelasticity described by the Boltzmann superposition principle, with relaxation or creep functions given analytically or numerically in the time or the frequency domain.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 301-303 
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    Notes: A new way of estimating critical damping in dynamic relaxation is proposed using the concept of a mass-dependant load vector, a postulate concerning vibrations and Rayleigh's principle. The method should be particularly appealing in those not uncommon circumstances where the actual loads are approximately proportional to the masses for then the computer time required in evaluating the various constants would actually be a part of the problem-solving time.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 303-304 
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    International Journal for Numerical Methods in Engineering 4 (1972) 
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 304-304 
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 331-336 
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    Notes: An iterative procedure of structural optimization is discussed that combines finite element analysis with the use of explicit optimality conditions. The procedure is illustrated by examples concerning sandwich beams with constraints on the deflection at a specified section, the maximum deflection or the sum of the maximum deflection in all spans of a continuous beam. The examples suggest that, starting from a uniform beam, by far the greatest saving in weight is obtained in the first two or three iterations.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 337-344 
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    Notes: The finite element technique is extended to an ideal gaseous mixture to study diffusion of several gaseous species in a carrier gas under various geometries and flow conditions. This is accomplished by obtaining the finite element analog of the balance of mass for each species, along with the corresponding equations for the balance of mass, momentum and energy for the mixture. Numerical results are obtained for a steady-state, incompressible, isothermal, viscous mixture of two gases for several boundary conditions. Favourable agreement is obtained between the finite element results and analytical solutions.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 452-452 
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 452-452 
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    International Journal for Numerical Methods in Engineering 5 (1972), S. 137-139 
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    Notes: It is shown that isoparametric finite elements with derivative degrees of freedom of second order or higher in the local curvilinear space demand an excessive number of degrees of freedom in the global space. To illustrate this limitation as it pertains to the continued development of higher order elements, discussion is centered around the general hexahedron element.
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    International Journal for Numerical Methods in Engineering 5 (1972), S. 141-142 
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    International Journal for Numerical Methods in Engineering 5 (1972), S. 113-135 
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    Notes: Several iterative processes for solving elasto-plastic problems are discussed in the context of a general formulation which includes: 1Associated and non-associated plasticity relations.2Strain hardening as well as strain softening behaviour.Advantages of the ‘initial stress’ process are emphasized for a wide category of problems and the isoparametric formulation is used throughout.
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    International Journal for Numerical Methods in Engineering 5 (1972), S. 142-145 
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    International Journal for Numerical Methods in Engineering 5 (1972), S. 145-147 
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    International Journal for Numerical Methods in Engineering 5 (1972) 
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    International Journal for Numerical Methods in Engineering 5 (1972), S. 148-148 
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    Notes: The constitutive equation is assumed in a very general form which includes as special cases non-linear creep, incremental elasto-plasticity as well as viscoelasticity represented by a chain of n standard solid models. Subdividing the structure into N finite elements, the problem of structural analysis is formulated with a system of 6N(n + 1) ordinary non-linear first-order differential equations in terms of the components of stresses and strains in the elements. This formulation enables one to apply Runge-Kutta methods or the predictor-corrector methods.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 17-22 
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    Notes: Some elements commonly used for analysis are examined for examined for completeness of polynomial interpolation and computational efficiency. Extensions to n-dimensional space are shown to be natural consequences of the interpolation, thus all elements considered here allow for finite element approximation in higher than three-dimensional spaces (e.g. space-time interpolations). From the study it is concluded that ‘serendipity’ class elements from the most efficient elements up to third-degree polynomial approximations. The method used here to develop the serendipity shape functions allows for different orders of interpolation along each edge. Thus, in zones where high accuracy is required meshes can now be easily changed from linear to quadratic or higher-order elements. Computations on some simple problems have demonstrated this to be a superior method than using large numbers of low ordered elements.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 49-65 
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    Notes: The application of the high precision triangular plate bending element to problems with curved boundaries is considered. Appropriate edge conditions for nodal points on these boundaries are derived. The error inherent in representing the shape of a curved boundary by a series of straight segments is found to be the limiting factor on accuracy, while the effect of approximations in the actual boundary conditions is minor. To overcome the first type of error, the high precision element is modified to include one curved edge. Substantial improvements in accuracy are obtained, as demonstrated in example calculations for circular and elliptical plates.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 85-93 
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    Notes: By formulating the potential boundary value problems in terms of physical (in contrast to purely mathematical) variables, the existence of dual energy principles may be clearly distinguished. In direct analogy with elasticity, two possible energy methods present themselves giving rise to two possible finite element procedures. General upper and lower bounds on an energy quantity may be obtained, which in the case of seepage become the total seepage rate for a certain class of seepage problem. For this case of seepage the derivation of the finite element procedure and the computing of the bounds is described.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 357-366 
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    Notes: This paper formulates the viscoelastic problem so as to allow the use of higher-order differential equation solution techniques. Viscoelastic models describing the functional relationships of stress and strain for the material are used. However, the procedure outline is valid for more general relationships. The advantages of using Runge-Kutta integration formulae are indicated and an example demonstrates the excellent results obtainable. The computational efficiency of the method is emphasized and error prediction is discussed.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 405-413 
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    Notes: It is shown that the Helmholtz equation in three dimension leads to finite element approximations on tetrahedral elements that closely resemble the corresponding two-dimensional treatment on triangle. For each polynomial order, there exist two numeric universal matrices independent to tetrahedron size and shape; the element matrices are always given as linear combinations of row and column permutations of these. Numeric matrices are given up to third-order, and the permutation schemes are shown in detail. Experimental computer programs using these elements have shown fast matrix assembly times; convergence rates are essentially similar to those obtained with the corresponding triangular elements.
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    Notes: A method is presented for the derivation of displacement fields which satisfy compatibility of normal slopes at inter-element boundaries and the condition of internal equilibrium. The displacement field is obtained by integrating the governing partial differential equation over the element, using the initial value approach. The method is applied to calculate the stiffness matrix of a triangular bending element with nine degrees of freedom. Comparison with published solutions based on various element models shows a high degree of accuracy and convergence of the method. The advantage of internal equilibrium is illustrated by an example involving slab-column interaction. The solution correlates satisfactorily with existing experimental results.
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    Notes: A generalized eigenvalue algorithm is presented herein along with the complete listing of the associated computer program, which may be conveniently utilized for the efficient solution of certain broad classes of eigenvalue problems. Extensive applications of the procedure are envisaged in the analysis of many important engineering problems, such as stability and natural frequency analysis of practical discrete structural systems, idealized by the finite element technique. The procedure based on the Sturm sequence method is accurate and fast, possessing several significant advantages over other known methods of such analysis. Numerical results are also presented for two representative structural engineering problems.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 415-432 
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    Notes: A finite element incremental stiffness formulation for determining finite displacement flexure behaviour of thin elastic plates is presented. The solution of the finite displacement problems is based on a piecewise linear incremental approach. The procedural aspects of the incremental approach, the control of incremental sizes and the accuracy of the incremental approximation are investigated. The details concerning how to obtain in-plane stresses and how to treat edge conditions are discussed. Results include rectangular plates under lateral pressure with various edge conditions and aspect ratios. The laterally loaded plate with edges movable in the plane of the plate is also treated.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 445-448 
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 433-444 
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    Notes: The problem of the elastica is generalized to include the effect if shear as well as axial strain. This leads to a non-linear two-point boundary-value problem composed of five first-order differential equations. At first the problem is linearized to study the way in which the bucking loads are modified by the combination of effects. For a simple column, the eigenvalues are found to be zeros of a third-order polynomial, and these are evaluated by a straightforward Newton-Raphson technique using a digital computer. The postbucking behaviour is then studied by using the computer and a ‘shooting’ method to solve the non-linear problem. For the special case of the prismatic column, graphs plotted by the computer are displayed to show the variation of load with both lateral and axial deflection for various combinations of axial compressibility and shear deformability. For one particular case, the deflected center line at various load levels is shown.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 451-452 
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 561-578 
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    Notes: The task of optimum design of structures is one that involves non-linear Programming. This is particularly the case when deflection as well as stress limitation are imposed as design criteria. In this paper a Fundamental approach is presented for the minimum weight design of statically indeterminate elastic structure subject to non-linear deflection and stress constrains.The matrix force method is adopted to formulate the problem. The constraints thus derived prove to have separable variables and therefore can be linearized in a piecewise manner. The computation aspects concerning the restricted entry in to the simplex tableau are briefly discussed and illustrative examples are given to verify the validity of the proposed method.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 587-596 
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    Notes: This paper describes a numerical method for solving first-order coupled matrix differential equations. Recursive equations are used to find ‘reflection’, ‘transmission’ and ‘source’ coefficients; these coefficients are then used to construct the vector solution to the differential equations. The method can be used to solve linear and non-linear differential equations with specified initial or two-point boundary values. Numerical results for several initial value problems are given in the paper.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 583-586 
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    Notes: This paper shows the importance of the rigid displacement condition when the finite element method is used for structures having little rigidity. The theoretical analysis is confirmed by numerical results obtained with a curved beam.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 579-582 
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    Notes: A procedure for the acceleration of convergence in non-linear analysis using the modified Newton-Raphson problem is described. The process has been found useful in plasticity and large deformation studies and its application elsewhere can be envisaged.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 597-599 
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 599-600 
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    International Journal for Numerical Methods in Engineering 5 (1972) 
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    International Journal for Numerical Methods in Engineering 5 (1972), S. 17-24 
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    Notes: This paper describes a very fast numerical method for solving the full dynamical equations governing the motion of the crankshaft and, as an example, the method is used to calculate the effect of clearance tolerance on the performance of a typical bearing. It was found that the minimum film thickness is not greatly affected by the clearance tolerance, suggesting that a fair degree of variation of clearance can be allowed.
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    International Journal for Numerical Methods in Engineering 5 (1972), S. 163-169 
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    Notes: In this paper finite element method for natural vibration problems, based on a modified Rayleigh-Ritz method is presented. Numerical results of a few simple examples indicate that the method is promising.
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    International Journal for Numerical Methods in Engineering 5 (1972), S. 151-162 
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    Keywords: Engineering ; Engineering General
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    Notes: A finite difference method for solving mixed initial and boundary value problems governed by hyperbolic partial differential equations is described. The method has been developed specifically to calculate the flow field associated with any arbitrary two-dimensional or axi-symmetric body placed in a uniform supersonic stream. In order to preserve accuracy in the neighborhood of the body surface, the equations governing the motion are formulated so that 1Streamlines form one system of co-ordinate curves.2Two of the equations of motion correspond to characteristic relations.The boundary condition at the body is applied by omitting the equation associated with the characteristic curve reaching the body surface from inside. The method presented is a pseudo-viscous method and flow fields which include shock waves can be treated with ease. The method has been used to calculate the flow past a particular axi-symmetric waisted body. The shape of this body is such that a secondary internal shock wave is generated by coalescence of Mach lines behind the bow shock wave. The results obtained are compared with theoretical and experimental results obtained by other workers. The investigation has been carried out as part of a programme of work needed to validate the pseudo-viscous method used.
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    International Journal for Numerical Methods in Engineering 4 (1972) 
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 155-161 
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    Notes: Two variations of an iterative scheme are presented for the solution to algebraic eigensystems. These algorithms are used in conjunction with the method of reduced generalized co-ordinates so that n number of frequencies and mode shapes are obtained simultaneously. The bases of the scheme are the well-known results of Stodola-Vianello and Gram-Schmidt. Advantages of the present schemes are realized both in computational effort and in computer storage. Examples are presented to illustrate the convergence characteristics.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 175-179 
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    Notes: Sources of data errors in the analysis of large frameworks are listed. Outputs of various forms for error detection are discussed; emphasis is given to the advantages of graphical output. For further error detection, a few procedures that can be included in computer programs are listed.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 181-193 
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    Notes: The finite element method is applied to the free transverse vibration of circular and annular plates of varying thickness. An annular element is derived which incorporates the number of diametral modes in the deflection function. This results in an element having only four degrees of freedom, these being the deflection and slopes at the inner and outer radii of the element at an anti-node of the particular vibration mode. Thickness variation in the radial direction is readily introduced, and stiffness and inertia matrices are presented for elements with linear and parabolic variations in thickness.The method is checked with several numerical examples. Calculations of free vibration of circular and annular plates of constant, linear and parabolic thickness variation are compared with available exact solutions.
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    Notes: An efficient digital computer method is presented for the determination of propagation of elastic stresses and deformations in certain geometrically non-linear structures subjected to high impact loading. The finite element matrix displacement approach utilizing curved quadrilateral shell elements in conjunction with a nodewise predictor-corrector method employing Runge-Kutta extrapolation techniques has been adopted for the present solution.The related computer program written in FORTRAN V for the UNIVAC 1108 computer has proved to be effective for the solution of a range of practical problems including rectangular and cylindrical panels. Numerical results are presented for a relevant structure, the cell container and the negative electrode of an impact-resistant battery subjected to high impact, simulating its free landing on a planetary surface.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 475-489 
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    Notes: Two FORTRAN IV subroutines that generate the stiffness, mass and geometric matrices for plate bending and plane stress elements using displacement models are described. The elements can be of either rectangular or triangular shape with anisotropic properties together with a linear variation in thickness. Although such a formulation is straightforward theoretically, emphasis has been placed on the programming technique where the amount of algebra involved in the matrix manipulations is avoided. Exact integration is automatically performed since a closed-form solution con be obtained.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 491-499 
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    Notes: Recent published research on structural optimization subject to dynamic constraints is reviewed, and suggestions for further work are offered. Both methods for handling constraints on natural frequencies and methods for treating the more difficult constraints on quantities directly related to the dynamic response are discussed. Computational aspects are emphasized throughout.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 455-473 
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    Notes: Plane stress structures of any shape and boundary conditions are simulated by finite element models with homogeneous stress and strain fields in each element. Besides the given live loads, dead loads (such as self-weight) depending on the unknown thickness distribution are allowed for. Possible practical requirements on geometry (minimum thickness, areas of equal thickness, areas of thickness variation in a prescribed way) are taken into account. Yield surfaces of the materials are piecewise linearized. On this basis the minimum weight design problem is formulated in terms of a linear program. This program is dualized and the pair of dual programming problems is discussed. The mechanical interpretation leads to the generalization of some known limit design theory results.Some numerical examples are given at the end.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 501-507 
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    Notes: The weighted residual and finite element methods are combined to yield a numerical solution scheme for the linerized BGK Boltzmann equation which describes the dynamic behaviour of gases in all flow régimes ranging from the free molecular to the continuum régime.The potential of the proposed method is illustrated by application to the problem of cylindrical Couette flow. Numerical results are compared with alternate numerical solutions and experimental data.
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    International Journal for Numerical Methods in Engineering 5 (1972), S. 1-16 
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    Notes: A ring type finite element model based on the discrete Kirchhoff hypothesis is developed and its feasibility for use in shell analysis is illustrated by determining the linear axisymmetric deformation behaviour of a conical shell loaded by lateral pressure. The displacement field is represented by a linear function of the meridional co-ordinate. The effects of coupling between bending and extension are included and in this respect the analysis is an extension of earlier work by other investigators using this same model, where bending deformations only were considered. The results of the investigation indicate that the proposed model is capable of accurately described coupled bending and extension effects with a relatively small number of elements. In addition to the numerical results obtained several distinct advantages and disadvantages of this element are brought out. In particular, the element stiffness coefficients can be easily derived because of the simplicity of the assumed vector field. However, it appears that the price paid for such a simple vector field approximation is an undue amount of machine storage, even for a relatively small number of elements. Several suggestions are made for minimizing some of these difficulties.
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    International Journal for Numerical Methods in Engineering 5 (1972), S. 253-270 
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    Notes: This paper is concerned with the development of a computational algorithm to find the optimal solution to the fluid network distribution design problem. The problem is formulated as a non-linear mathematical optimization problem and is solved using a sequential unconstrained minimization technique. Illustrative examples are included and the design solutions obtained compared with the solutions obtained using a parametric method.
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    International Journal for Numerical Methods in Engineering 5 (1972), S. 243-252 
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    Notes: The accuracy and convergence properties of two new Timoshenko beam finite elements are compared to those now widely used. Both of the new elements are consistently derived from a variational principle written in terms of the transverse beam deflection and normal rotation. The deflection is expanded as a cubic and the rotation as a quadratic in the axial co-ordinate of the beam, giving rise to a seven degrees of freedom element. The first element, called TIM7, includes all the degrees of freedom while the second, called TIM4, reduces this number to four through a constraint introduced by Egle. Both elements display monotonic covergence for beam natural frequencies as a function of the number of degrees of freedom and both will give the analyst some measure of confidence in response calculations for beams excited at high frequencies.
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    International Journal for Numerical Methods in Engineering 5 (1972), S. 271-275 
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    Notes: A direct numerical method is presented for the solution of a class of non-linear integro-differential equations which arise in the finite deflection of thin viscoelastic beams. The method is not iterative and thus avoids the convergence problems often associated with iterative methods. The analysis and results are given for the deflection of a uniformly loaded cantilever.
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    International Journal for Numerical Methods in Engineering 5 (1972), S. 277-288 
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    Notes: Two general quadrilateral elements for plate bending are developed. Each has 12 degrees of freedom and accounts for transverse shear deformation effects. Each is built from four triangular elements whose properties are derived by the assumed-stress hybrid approach. Matrices needed to generate the stiffness properties of the triangular elements are explicitly stated so as to facilitate their use. Numerical test cases, in which shear deformation effects range from negligible to very important, are used to illustrate the behaviour of the elements. It is concluded that the simpler of the two quadrilaterals is one of the best plate elements currently available.
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    International Journal for Numerical Methods in Engineering 5 (1972), S. 289-296 
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    Notes: An improved finite difference method is presented. This is exemplified by computing the natural frequency of vibrating plates. A new formulation of the problem showing computational advantages is derived. A procedure for efficient implementation of the improved method is given. Numerical examples are calculated showing results comparable to other methods but employing far fewer unknowns.
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    International Journal for Numerical Methods in Engineering 5 (1972), S. 297-300 
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    Notes: Explicit formulae for the density of a class of finite element matrices are presented. The formulae are functions of the characteristics of the mesh and the element (piecewise polynomial) employed. Results are given for two-dimensional triangular and quadrilateral elements and three-dimensional tetrahedral and cuboid elements. These results are useful in the management of storage in finite element computer programmes.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 23-37 
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    Notes: A computer program for surge tank analysis, which gives a complete history of the water level in the tank and the fluid velocity in the pressure conduit from the time of initiation of value closure, is described and presented. The governing equations are solved by third-order Runge-Kutta procedure and losses in head in the pressure conduit and surge tank constriction. A major aspect in the paper is the non-dimensional from of the governing parameters, facilitating the use of the program in preliminary design situations.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 39-47 
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    Notes: A numerical method for the direct computation of the particle densities, mass density, enthalpy and specific heat at constant pressure of a monatomic ionized gas in thermodynamic equilibrium is presented. This method permits calculation of the thermodynamic properties without iteration or numerical differentiation of any of the formulae.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 109-122 
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    Notes: A mathematical model of the two-dimensional, incompressible turbulent boundary layer in arbitrary pressure gradient described and solved in earlier work1 is solved by an implicit finite difference method. Comparisons of skin friction coefficient and momentum thickness values for the two methods for four test cases of different character indicate that the present method is a useful alternative to the method of characteristics which was used in previous work.One feature of the present method is that allowance is made for the logarithmic character of the velocity profile when constructing one of the finite difference expressions.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 123-132 
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    Notes: The finite element representation of the linearized equations governing the steady compressible flow of an isentropic perfect gas is considered. The fully non-linear system of equations is linearized on the basis of small perturbation theory. The finite element matrix equations for an arbitrary polygonal element are generated by the method of weighted residuals: Galerkin's criterion is used. As an example, a triangular element in two-dimensional flow is treated in detail and numerical result for a sample problem are given.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 95-107 
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    Notes: This paper deals with the stability of the numerical solutions of a dynamic finite element analysis. The solutions are obtained through a stepwise integration of the equations of motion. Upper bounds on the steplength of the integration are obtained from a stability analysis of using a simple finite difference approximation for the equations of motion, and are shown to depend strongly on the particular element is use and on how the mass of the element is distributed at its nodes.As an example, the two-dimensional wave propagation in a semi-infinite plate subjected to a suddenly applied moment along its edge is studies. Through the example, we show that the bound on the steplength, obtained from the simple analysis, can provide a useful guide on choosing the steplength in other higher order integration methods. In particular, we show that, for stability considerations, the upper bound on the steplength should also hold for a fourth order explicit method. In order to achieve an acceptable accuracy of the solution, we show that the steplength should be approximately one half of the bound for the higher order explicit method as well as a higher order implicit method.Solution of the example has been compared with that of the Timoshenko theory.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 345-355 
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    Notes: A method is proposed for the inclusion of stochastic parameters, using a Monte Carlo simulation technique, in the lower bound plastic analysis of a structural continuum by non-linear mathematical programming. The particular example chosen is a circular, cylindrical concrete silo subject to loading by its contents.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 307-329 
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    Keywords: Engineering ; Engineering General
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    Notes: The inelastic flexural behaviour of reinforced concrete beams is described, as usual, by a trilinear (or, in general, multilinear) moment-rotation relationship. The analysis of beams and frames subject to given external actions is formulated as a ‘linear complementary problem’ and, in two dual ways, as a ‘quadratic programming problem’. These problems are solved by means of classical and recent algorithms in use in operations research. By the procedures pointed out it is also possible to calculate the distribution of plastic deformations along the beams. Two methods are proposed for the evaluation of the safety factor against local failure due to limited rotation capacity: the solution technique of the direct one essentially consists of an application of the simplex method for linear programming. Finally, a procedure is established for determining, allowing for the irreversibility of plastic rotations, the structural response to non-proportional loading histories which consist of individually proportional stages. Illustrative examples are given.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 509-522 
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    Notes: This paper presents a method of numerical solution of boundary value problems governed by a set of ordinary differential equations. The highest derivative is chosen as the unknown function. Governing equations are transformed into a set of integral equations. The kernels of integral equations turn out to be influence functions for deflection and/or bending moments of a corresponding beam, and can, therefore, be computed using well-known methods of structural analysis. Finally, the unknown function (highest derivative) is approximated in a defined manner and the solution is obtained through numerical integration.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 541-550 
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    Notes: The convergence and accuracy characteristics of two commonly used finite difference schemes, namely the central and the upwind difference, are compared with a scheme recently proposed by Spalding. For a simple two-dimensional problem with conduction and convection, it is shown that the scheme proposed by Spalding is preferable choice both from the point of view of accuracy and that of the convergence. The extension of the conclusion to a general case must await further substantiation.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 523-539 
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    Notes: The method presented in Part I of this paper1 is extended to problems governed by a system of partial differential equations. The basic clarity and simplicity of the method is preserved.The domain of integration is covered by a net of orthogonal (not necessarily equidistant) straight lines. Choosing again highest derivatives with respect to each of the co-ordinates as unknowns, it is possible to reduce the problem to a system of integral equations along the lines of the adopted net.The subsequent procedure is basically the same as in Part I. The results prove to be of remarkable accuracy even for very coarse nets.
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    International Journal for Numerical Methods in Engineering 4 (1972), S. 551-559 
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    Notes: It is shown that the upwind difference scheme of formulating differential expressions, in problems involving transport by simultaneous convection and diffusion, is superior to the central differences scheme, when the local Peclet number of the grid is large. Even better schemes are derived and discussed. It is pointed out that the best finite differences analogues are found by approximating differential expressions as a whole, and that simple (e.g. one-dimensional) exact solutions form a useful, legitimate and independent source of these optimum algebraic formulae.
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    International Journal for Numerical Methods in Engineering 5 (1972), S. 25-39 
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    Notes: A finite difference method for analyzing two-dimensional steady-state potential flow in a zoned anisotropic material is described.A triangular mesh is used is as to avoid difficulties often occurring in the more familiar rectangular mesh where fractional mesh spaces exist at irregular boundaries. The system of finite difference equations based on the triangular mesh is assembled in a form such that the coefficient matrix is 3-cylic and consistently ordered.1, 2 This enables the optimum accelerating factor for successive overrelaxation to be closely estimated. Since the rate of convergence to the solution, and consequently the efficiency of the method, increases rapidly as the accelerating factor tends towards its optimum value, close determination of the optimum is important. The method is particularly suitable for problems having large numbers of nodes.Examples of the method on specific problems are given. The solution accuracy is assessed and has been found acceptable.
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    International Journal for Numerical Methods in Engineering 5 (1972), S. 41-55 
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    Notes: The finite element method is applied to the solution f seepage problems where both electrokinetic and hydrodynamic forces occur. The resulting system of coupled equations is used to solve both the steady state and transient conditions in a one-dimensional system and compared with the theoretical result. An analysis is then made of a more complex two-dimensional problem where the application of electro-osmosis may be used successfully to prevent piping.
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    International Journal for Numerical Methods in Engineering 5 (1972), S. 57-64 
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    Keywords: Engineering ; Engineering General
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    Notes: Numerical calculation of the collapse load of a reinforced concrete plate is considered. Mathematical programming formulations are derived by using finite element representation. The plate is assumed to obey the square yield criteria. It is shown that good lower bound solutions can be obtained on a systematic basis. Upper bound solutions are approximated result of yield-line theory. An example of the square built-in plate loaded uniformly is given.
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    Type of Medium: Electronic Resource
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  • 96
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    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 5 (1972), S. 65-82 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: An alternative approach to the usual finite element treatment of steady-state temperature problems is presented, using approximations for the field of the dual variables.The appropriate extremum principle is established and its minimization is discussed in connection with a plane triangular finite element process. Original heat flow elements are derived: in conjunction with temperature elements, they enable dual analysis of a given structure and an important estimate of the convergence to the true solution by upper and lower bounds to the dissipation function, as illustrated by means of several examples.
    Additional Material: 6 Ill.
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  • 97
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    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 5 (1972), S. 83-89 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A method of formulating and computing numerically integrated stiffness matrices is presented. Through a better organization of the formulation which is easily achieved by the use of index notation, the computations for integrating and evaluating the element stiffness matrix are substantially reduced. For cases in which the terms of the material property tensor are not functions of position co-ordinates (elastic isotropic and anisotropic materials), a further reduction in computation is achieved by multiplying the material property-tensor after the numerical integration has been carried out.
    Additional Material: 1 Ill.
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  • 98
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    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 5 (1972), S. 91-101 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Relations form the theory of characteristics are used to connect boundary and interior points in the numerical analysis of multidimensional, unsteady gas dynamics.Since specification of all variables on the boundaries does not occur, it is necessary to employ at least some of the differential equations for boundary points. The objective is to avoid the use of non-central finite differences and the subsequent generation of a pseudo-boundary layer.Boundaries consisting of a moving shock wave, a solid wall and a surface in a supersonic, downstream region are considered. The numerical procedure for determining variables on boundaries is described in datail for the hypersonic, blunt body problem.
    Additional Material: 5 Ill.
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  • 99
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    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 5 (1972), S. 103-112 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Lake circulations around island are calculated based on the shallow lake model using the finite element method. Wind stress, bottom friction, Coriolis force and shore line geometry are incorporated in the mathematical model and Lake Erie is used in the computations. A closure condition is derived for the multiply connected region of interest. The discussions are made in the light of accuracy and applicability of the numerical scheme used, and the influence of the circulation pattern due to the presence of islands.
    Additional Material: 8 Ill.
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  • 100
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    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 5 (1972), S. 147-148 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Type of Medium: Electronic Resource
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