Integration of Area Coordinates in Matrix Structural Anallysis

Stricklin, J. A.

doi:10.2514/3.55425

Cited by 12 publications

(3 citation statements)

References 10 publications

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“…For satisfying (2) while keeping the system natural to the element, we further assume that the pth vertex is described as i = 1,2, ...,n (4) where di p is Kronecker's symbol.…”

Section: Linear Coordinate Systemmentioning

confidence: 99%

Some aspects of the natural coordinate system in the finite-element method.

Fried

1969

AIAA Journal

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by numerically solving the time-dependent equations of motion (hyperbolic in form) in the subsonic, transonic, and supersonic flow regions. The calculation was carried out by Prozan 13 and information on the technique is described by Saunders. 14 Starting from an initially prescribed flowfield taken to be one-dimensional, the steady-state solution that is sought is obtained as a limit of the solution of the nonsteady problem. Boundary conditions must be prescribed around the entire flowfield, and this requires an a priori specification of the mass flow rate, a quantity that actually, however, is to be calculated. This prediction yields an inviscid flow coefficient of 0.990, which is higher than the experimentally measured value of 0.985; but the resulting Mach number distribution agrees generally with the experimental results in the transonic region (Fig. 2). Of particular interest is the similarity between the experimental (Fig. 1) and the predicted (Fig. 2) Mach number contours in the shock-formation region. Along the wall the predicted values (Fig. 3) are somewhat low just downstream of the sonic condition but agree with the data farther downstream, where the Mach number peaks and then decreases slightly. The predicted decrease in the Mach number (wall pressure rise) just downstream of the tangency, however, is larger than that determined experimentally. The pressure rise is important since, under certain conditions, this adverse pressure gradient can significantly influence the boundary layer and, thus, the heat transfer from high-temperature gas flows. 5 Prozan's prediction agrees well with the experimental data along the centerline.In two method-of-characteristics predictions shown in Fig. 3, good agreement is found with the experimental results, and a slight Mach number decrease is predicted approximately where observed. Kliegel used a variableMach-number start line that varied from M = 1.3 along the wall to M = 1.1 along the centerline; Shelton used a start line of M = 1.3 and the program of Butler. 15 Summary and ConclusionsInternal flow measurements in a nozzle with a small ratio of r c /rth of 0.625 revealed radial variations in the flow. At the physical throat the Mach number was 0.8 at the axis and 1.4 near the wall. The three methods considered predict the transonic flowfield reasonably well in nozzles which have a small throat radius of curvature (r c /r t h < 1.0), such as that for the nozzle investigated.

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“…For satisfying (2) while keeping the system natural to the element, we further assume that the pth vertex is described as i = 1,2, ...,n (4) where di p is Kronecker's symbol.…”

Section: Linear Coordinate Systemmentioning

confidence: 99%

Some aspects of the natural coordinate system in the finite-element method.

Fried

1969

AIAA Journal

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“…for every P that satisfies (46). By virtue of the fundamenta lemma of the calculus of variations, P then satisfies (9) and (11) In addition, since P eF, boundary condition (10) is also satisfied.…”

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confidence: 99%

“…Mechanical Development Department, General Motors Research Laboratories, Warren, Mich.6 Numbers[11][12][13][14][15][16][17] in brackets designate Additional References at end of closure.…”

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confidence: 99%

Discussion: “Finite Element Solution of the Steady-State Compressible Lubrication Problem” (Reddi, M. M., and Chu, T. Y., 1970, ASME J. Lubr. Technol., 92, pp. 495–502)

Booker

1970

Journal of Lubrication Technology

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On finite element integration in natural co‐ordinates

Eisenberg

Malvern

1973

Numerical Meth Engineering

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Integration formulas for polynomial expressions in terms of natural co-ordinates are extensively quoted without proof in standard references on finite element theory. A derivation of these relations is presented and their application is extended to non-integral exponents. SUMMARYNumerical solutions for limit loads of pressurized cylindrical cantilever shells are given for the It is suggested that the numerical approach can be used for more complex problems for which the exact Il'yushin yield surface. exact solution based on equation (2) has not been obtained to date.

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Integration of Area Coordinates in Matrix Structural Anallysis

Cited by 12 publications

References 10 publications

Some aspects of the natural coordinate system in the finite-element method.

Some aspects of the natural coordinate system in the finite-element method.

Discussion: “Finite Element Solution of the Steady-State Compressible Lubrication Problem” (Reddi, M. M., and Chu, T. Y., 1970, ASME J. Lubr. Technol., 92, pp. 495–502)

On finite element integration in natural co‐ordinates

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