Browsing by Author "Wang, Chang Y."

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  • Convergence Theory of Probability-one Homotopies for Model Order Reduction
    Wang, Chang Y.; Bernstein, Dennis S.; Watson, Layne T. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1996-06-01)
    The optimal H-square model reduction problem is an inherently nonconvex problem and thus provides a nontrivial computational challenge. This paper systematically examines the requirements of probability-one homotopy methods to guarantee global convergence. Homotopy algorithms for nonlinear systems of equations construct a continuous family of systems, and solve the given system by tracking the continuous curve of solutions to the family. The main emphasis is on guaranteeing transversality for several homotopy maps based upon the pseudogramian formulation of the optimal projection equations and variations based upon canonical forms. These results are essential to the probability-one homotopy approach by guaranteeing good numerical properties in the computation- al implementation of the homotopy algorithms.
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    Deceleration of a Rotating Disk in a Viscous Fluid
    Watson, Layne T.; Wang, Chang Y. (AIP Publishing, 1979)
    A disk rotating in a viscous fluid decelerates with an angular velocity inversely proportional to time. It is found that the unsteady Navier–Stokes equations admit similarity solutions which depend on a nondimensional parameter S =α/Ω0, measuring unsteadiness. The resulting set of nonlinear ordinary differential equations is then integrated numerically. The special case of S =−1.606 699 corresponds to the decay of rotation of a free, massless disk in a viscous fluid.
  • Effect of a Sawtooth Boundary on Couette Flow
    Mateescu, Gabriel; Wang, Chang Y.; Ribbens, Calvin J.; Watson, Layne T. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1998)
    The relative tagential motion of a smooth plate and a corrugated plate separated by a viscous fluid is studied. The full Navier-Stokes equations are solved using Hermite collocation and Newton's method. Detailed streamlines and vorticity distributions are determined. The increased drag due to corrugation is found to be substantial.
  • Effect of Elevated Mass Center on the Global Stability of a Solid Supported by Elastica Columns
    Thacker, William I.; Wang, Chang Y.; Watson, Layne T. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1996)
    Recently we advocated a new stability index, the global critical load, for very elastic structures. This index is extremely useful for flexible structures under large disturbances such as earthquakes. The present note determines this index for a two-dimensional rigid solid supported by two flexible columns. Using the nonlinear elastica equations the buckling and postbuckling problem is solved by a homotopy nonlinear system solver. The present results show the bifurcation curve is quite sensitive to the elevated mass center. The global buckling load is drastically reduced although the critical buckling load of linear stability analysis is the same. An explanation is given through the study of a solid supported by one column.
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    Effect of flexible joints on the stability and large deflections of a triangular frame
    Thacker, William I.; Wang, Chang Y.; Watson, Layne T. (Department of Computer Science, Virginia Polytechnic Institute & State University, 2007)
    An isosceles triangular frame with rotationally resistive joints under a tip load is studied. The large in-plane deformation elastica equations are formulated. Stability analysis shows the frame can buckle symmetrically or asymmetrically. Post-buckling behavior showing limit load and hysteresis are obtained by shooting and homotopy numerical algorithms. The behavior of a frame with rigid joints is studied in detail. The effects of joint spring constant and base length are found.
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    The Equilibrium States of a Heavy Rotating Column
    Watson, Layne T.; Wang, Chang Y. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1982)
    A heavy, rotating vertical column is clamped at one end and free at the other end. The stability boundaries are found by both analytical approximations and numerical integration. The problem depends on two non-dimensional parameters: beta representing the importance of gravity to rigidity and alpha representing the importance of rotation to rigidity. Buckled shapes for the different modes are also obtained.
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    Free Rotation of a Circular Ring with an Unbalanced Mass
    Watson, Layne T.; Wang, Chang Y. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1988-05-01)
    Large space structures are much more flexible than their terrestrial counterparts. Rotation of large space structures may be desirable for stability, thermal or artificial gravity reasons. Previous literature includes the large deformations due to the free rotation of a slender rod and a ring about a diameter. An important model of a space station is a ring rotating about its axis of symmetry. If the ring is balanced it will be stable and remain circular. The present note considers the case when the ring is unbalanced by a mass attached to a point on the ring.
  • The Global Stability of a Rigid Solid Supported by Elastic Columns
    Thacker, William I.; Wang, Chang Y.; Watson, Layne T. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1994)
    A new stability index, the global critical load, is advocated. This index is useful for flexible structures prone to large disturbances such as earthquakes. A symmetric rigid body supported by flexible legs is studied in detail. The nonlinear equilibrium equations are solved and the results show that global stability depends heavily on the height of the mass center and the distance between the legs.
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    Hanging an Elastic Ring
    Watson, Layne T.; Wang, Chang Y. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1980)
    A thin flexible elastic circular ring is hung at one point. The ring deforms due to its own weight. The problem depends on a non-dimensional parameter B representing the relative importance of density and length to rigidity. The heavy elastica equations are solved by perturbation for small B, by a quasi-Newton method for intermediate B, and by a homotopy method for large B. The approximate results show good agreement with numerical integration for B < 20.
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    A Homotopy Method Applied to Elastica Problems
    Watson, Layne T.; Wang, Chang Y. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1980)
    The inverse problem in nonlinear (incompressible) elastica theory, where the end positions and inclinations rather than the forces and moment are specified, is considered. Based on the globally convergent Chow-Yorke algorithm, a new homotopy method which is simple, accurate, stable, and efficient is developed. For comparison, numerical results of some other simple approaches (e.g., Newton's method based on shooting or finite differences, standard embedding) are presented. The new homotopy method does not require a good initial estimate, and is guaranteed to have no singular points. The homotopy method is applied to the problem of a circular elastica ring subjected to N symnetrical point loads, and numerical results are given for N = 2,3,4.
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    Large Deformations of a Whirling Elastic Cable
    Wang, Chang Y.; Watson, Layne T. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1990)
    The large deformations of a whirling elastic cable is studied. The ends of the cable are hinged but otherwise free to translate along the rotational axis. The nonlinear governing equations depend on a rotation-elasticity parameter J. Bifurcation about the straight, axially rotating case occurs when J is greater than or equal to n(pi). Perturbation solutions about the bifurcation points and matched asymptotic solutions for large J are found to second order. Exact numerical solutions are obtained using quasi-Newton and homotopy methods.
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    Large Deformations of Rotating Polygonal Space Structures
    Watson, Layne T.; Wang, Chang Y. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1990)
    The effects of rotation on space structures are very important nowadays. Because the structures are large and thus relatively flexible, unacceptable stresses or deformations may occur due to centrifugal forces. In a recent paper we considered the effects of rotation on an unbalanced circular ring. The present note studies a related problem, i.e., the free rotation of polygonal frames. These shapes are basic in structural design.
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    The Nonlinear Stability of a Heavy Rigid Plate Supported by Flexible Columns
    Thacker, William I.; Wang, Chang Y.; Watson, Layne T. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1992)
    A heavy rigid platform is supported by thin elastic legs. The governing equations for large deformations are formulated and solved numerically by homotopy and quasi-Newton methods. Nonlinear phenomena such as non-uniqueness, catastrophe and hysteresis are found. A global critical load for nonlinear stability is introduced.
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    Overhang of a Heavy Elastic Sheet
    Watson, Layne T.; Wang, Chang Y. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1981)
    A flexible elastic sheet overhangs from a corner. The deflection due to its own weight depends on a parameter K which represents the relative importance of overhang length to the bending length (EI/p)^1/3.
  • Probability-one Homotopy Algorithms for Solving the Coupled Lyapunov Equations Arising in Reduced-Order H^2/H^(infinity) Modeling, Estimation, and Control
    Wang, Chang Y.; Bernstein, Dennis S.; Watson, Layne T. (Department of Computer Science, Virginia Polytechnic Institute & State University, 2000-04-01)
    Optimal reduced order modeling, estimation, and control with respect to combined H^2/H^(infinity) criteria give rise to coupled Lyapunov and Riccati equations. To develop reliable numerical algorithms for these problems this paper focuses on the coupled Lyapunov equations which appear as a subset of the synthesis equations. In particular, this paper systematically examines the requirements of probability-one homotopy algorithms to guarantee global convergence. Homotopy algorithms for nonlinear systems of equations construct a continuous family of systems and solve the given system by tracking the continuous curve of solutions to the family. The main emphasis is on guaranteeing transversality for several homotopy maps based upon the pseudogramian formulation of the coupled Lyapunov equations and variations based upon canonical forms. These results are essential to the probability-one homotopy approach by guaranteeing good numerical properties in the computational implementation of the homotopy algorithms.
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    Rotation of Polygonal Space Structures
    Wang, Chang Y.; Watson, Layne T. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1990)
    The free rotation in space of flexible polygonal structures about their axes is studied. The large deformation equations are formulated, resulting in a set of sixth order nonlinear differential equations. The governing parameter B represents the relative importance of centrifugal forces to flexural rigidity. Perturbation solutions are obtained for small B. The equations are also integrated numerically by quasi-Newton and homotopy methods. Forces, moments, and maximum deformations are determined for polygons of three to six sides and B up to 100. At very large B, the polygons deform into a circle.
  • Stability and Postbuckling of a Platform with Flexible Legs Resting on a Slippery Surface
    Thacker, William I.; Wang, Chang Y.; Watson, Layne T. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1997-12-01)
    A rigid platform is supported by thin elastic legs. The legs are able to slide on the ground as they deform. The governing equations for large deformations are formulated and solved numerically by homotopy and quasi-Newton methods. Nonlinear phenomena such as nonuniqueness are found. A global critical load for nonlinear stability is presented.
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    Steady Viscous Flow in a Trapezoidal Cavity
    McQuain, William D.; Ribbens, Calvin J.; Wang, Chang Y.; Watson, Layne T. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1992)
    The flow in a trapezoidal cavity (including the rectangular and triangular cavities) with one moving wall is studied numerically by finite differences with special treatment in the corners. It is found that streamlines and vorticity distributions are sensitive to geometric changes. The mean square law for core vorticity is valid for the rectangle but ceases to be valid for the triangular cavity.
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    Steady Viscous Flow in a Triangular Cavity
    Ribbens, Calvin J.; Watson, Layne T.; Wang, Chang Y. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1992)
    Steady recirculating viscous flow inside an equilateral triangular cavity is generated by translating one side. The Navier-Stokes equations are solved numerically using finite difference on a transformed geometry. The results show a primary eddy and a series of secondary eddies at the stagnant corner. For high Reynolds numbers the interior of the primary eddy has constant vorticity, but its value cannot be predicted by the mean-squared law.
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    Vorticity Induced by a Moving Elliptic Belt
    Ribbens, Calvin J.; Wang, Chang Y.; Watson, Layne T.; Alexander, Kevin A. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1989)
    The viscous flow inside an elliptic moving belt is studied using Newton's method on a Hermite collocation approximation. The streamlines and especially the vorticity distribution are found for Reynolds number up to 1000 and aspect ratio up to five. For low Reynolds numbers vorticity diffuses from regions of high curvature. For high Reynolds numbers there exists a closed boundary layer and a core of constant vorticity. The core vorticity compares well with the estimation from the mean square law.