Large-scale computation of incompressible viscous flow by least-squares finite element method

Jiang, Bo‐Nan; Lin, Tsung-Liang; Povinelli, Louis A.

doi:10.1016/0045-7825(94)90172-4

Cited by 160 publications

(159 citation statements)

References 28 publications

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“…Recently, there has been substantial interest in the use of least squares principles for the approximate solution of the Navier-Stokes equations of incompressible flow; for some examples of bona fide least squares methods, one may consult, e.g., [5,8,9,11,12,19,20,21,22,23,24,28]. The computational results provided in these papers indicate that the methods considered are effective; however, careful analyses of these methods indicate that they do not yield optimally accurate approximations.…”

Section: Introductionmentioning

confidence: 98%

“…If all equation indices are equal, one has the common and unimportant factor h2s' and, insofar as minimization is concerned, the functional (34) is identical to the standard one (21). Using standard techniques of the calculus of variations, one can show, for any fixed value of h , that minimization of (34) over an appropriate space U is equivalent to the variational problem For the discretization of (35) we consider a finite-dimensional subspace UA of U and pose the problem:…”

Section: The Mesh-dependent Least Squares Principlementioning

confidence: 99%

“…(The latter assumption may require additional information of the type (7).) Then, we define the standard least squares functional for the problem (2)- (3) by (21) *{U) = l|curlw + &adp " fll|(-*. .-*)+ Hcurlu " oe * M-* + \\divu-fi\\2_Si.…”

Section: Jamentioning

confidence: 99%

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Analysis of least squares finite element methods for the Stokes equations

Bochev¹,

Gunzburger²

1994

Math. Comp.

122

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Abstract. In this paper we consider the application of least squares principles to the approximate solution of the Stokes equations cast into a first-order velocity-vorticity-pressure system. Among the most attractive features of the resulting methods are that the choice of approximating spaces is not subject to the LBB condition and a single continuous piecewise polynomial space can be used for the approximation of all unknowns, that the resulting discretized problems involve only symmetric, positive definite systems of algebraic equations, that no artificial boundary conditions for the vorticity need be devised, and that accurate approximations are obtained for all variables, including the vorticity. Here we study two classes of least squares methods for the velocity-vorticitypressure equations. The first one uses norms prescribed by the a priori estimates of Agmon, Doughs, and Nirenberg and can be analyzed in a completely standard manner. However, conforming discretizations of these methods require C1 continuity of the finite element spaces, thus negating the advantages of the velocity-vorticity-pressure formulation. The second class uses weighted L2-norms of the residuals to circumvent this flaw. For properly chosen meshdependent weights, it is shown that the approximations to the solutions of the Stokes equations are of optimal order. The results of some computational experiments are also provided; these illustrate, among other things, the necessity of introducing the weights.

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Section: Introductionmentioning

confidence: 98%

Section: The Mesh-dependent Least Squares Principlementioning

confidence: 99%

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Analysis of least squares finite element methods for the Stokes equations

Bochev¹,

Gunzburger²

1994

Math. Comp.

122

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“…We tested the method on the lid-driven cavity represented by the unit square Ω = [10] for dealing with these singularities, the domain was uniformly partitioned into an N by N grid of squares, and then the top layer was further partitioned into a layer of height 75/N and a layer of height 25/N as the new topmost layer. Each rectangular cell was then partitioned into a pair of triangles.…”

Section: Matlab Direct Solver Methodsmentioning

confidence: 99%

“…The usual procedure for least squares treatment of a first-order system of nonlinear equations is to linearize the equations by a Newton iteration or fixed-point iteration, and solve a linear least squares problem at each step; see, e.g., [10]. If a Newton iteration is used, the method is equivalent to applying a Gauss-Newton method to find a zero of the gradient of the nonlinear functional associated with the sum of squared residuals.…”

Section: Nonlinear Least Squares Functionalmentioning

confidence: 99%

A Sobolev gradient method for treating the steady-state incompressible Navier-Stokes equations

Renka

2013

Open Mathematics

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Abstract:The velocity-vorticity-pressure formulation of the steady-state incompressible Navier-Stokes equations in two dimensions is cast as a nonlinear least squares problem in which the functional is a weighted sum of squared residuals. A finite element discretization of the functional is minimized by a trust-region method in which the trustregion radius is defined by a Sobolev norm and the trust-region subproblems are solved by a dogleg method. Numerical test results show the method to be effective. MSC:65N30

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Three-Dimensional Transient Mold Cooling Analysis Based on Galerkin Finite Element Formulation With a Matrix-Free Conjugate Gradient Technique

Tang

Pochiraju

Chassapis

et al. 1996

Int. J. Numer. Meth. Engng.

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SUMMARYA methodology is presented to simulate the three-dimensional heat transfer within a mold during the injection molding process. The mold cooling analysis assists cooling channel design and paves the way for part shrinkage and warpage analysis. The transient temperature distributions in the mold and the polymer part are simultaneously computed by Galerkin Finite Element Method (GFEM) using a matrix-free Jacobi Conjugate Gradient (JCG) scheme. The numerical method presented here is efficient and has shown to require a fraction of the memory and computing time required by conventional methods. The matrix-free algorithm is initially validated using an injection mold designed to produce a plaque with a molded-in hole. Subsequently, the method is further applied to a representative automotive plastic component.

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Large-scale computation of incompressible viscous flow by least-squares finite element method

Cited by 160 publications

References 28 publications

Analysis of least squares finite element methods for the Stokes equations

Analysis of least squares finite element methods for the Stokes equations

A Sobolev gradient method for treating the steady-state incompressible Navier-Stokes equations

Three-Dimensional Transient Mold Cooling Analysis Based on Galerkin Finite Element Formulation With a Matrix-Free Conjugate Gradient Technique

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