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Projection method

About: Projection method is a(n) research topic. Over the lifetime, 3685 publication(s) have been published within this topic receiving 85244 citation(s).
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Journal ArticleDOI
TL;DR: An algorithm for solving large nonlinear optimization problems with simple bounds is described, based on the gradient projection method and uses a limited memory BFGS matrix to approximate the Hessian of the objective function.
Abstract: An algorithm for solving large nonlinear optimization problems with simple bounds is described. It is based on the gradient projection method and uses a limited memory BFGS matrix to approximate the Hessian of the objective function. It is shown how to take advantage of the form of the limited memory approximation to implement the algorithm efficiently. The results of numerical tests on a set of large problems are reported.

4,483 citations


Journal ArticleDOI
Mark Sussman1, Peter Smereka1, Stanley Osher1Institutions (1)
TL;DR: A level set method for capturing the interface between two fluids is combined with a variable density projection method to allow for computation of two-phase flow where the interface can merge/break and the flow can have a high Reynolds number.
Abstract: A level set approach for computing solutions to incompressible two-phase flow is presented. The interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function. A new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface. We never have to explicitly reconstruct or find the zero level set. Consequently, we are able to handle arbitrarily complex topologies, large density and viscosity ratios, and surface tension, on relatively coarse grids. We use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations. We consider the motion of air bubbles and water drops. We also compute flows with multiple fluids such as air, oil, and water.

3,815 citations


01 Jun 1995-
Abstract: A level set method for capturing the interface between two fluids is combined with a variable density projection method to allow for computation of two-phase flow where the interface can merge/break and the flow can have a high Reynolds number. A distance function formulation of the level set method enables one to compute flows with large density ratios (1000/1) and flows that are surface tension driven; with no emotional involvement. Recent work has improved the accuracy of the distance function formulation and the accuracy of the advection scheme. We compute flows involving air bubbles and water drops, to name a few. We validate our code against experiments and theory.

3,556 citations


Journal ArticleDOI
TL;DR: This paper presents a class of TVB (total variation bounded) discontinuous Galerkin finite element methods for solving conservation laws ut+Σi=1d(fi(u)xi=0.1d) using a 1-dimensional system as a model, and discusses different implementation techniques and theories analogous to scalar cases proven for linear systems.
Abstract: This is the third paper in a series in which we construct and analyze a class of TVB (total variation bounded) discontinuous Galerkin finite element methods for solving conservation laws ut+Σi=1d(fi(u)xi=0. In this paper we present the method in a system of equations, stressing the point of how to use the weak form in the component spaces, but to use the local projection limiting in the characteristic fields, and how to implement boundary conditions. A 1-dimensional system is thus chosen as a model. Different implementation techniques are discussed, theories analogous to scalar cases are proven for linear systems, and numerical results are given illustrating the method on nonlinear systems. Discussions of handling complicated geometries via adaptive triangle elements will appear in future papers.

1,309 citations


Journal ArticleDOI
Abstract: In this paper we describe a second-order projection method for the time-dependent, incom­ pressible Navier-Stokes equations. As in the original projection method developed by Chorin, we first solve diffusion-convection equations to predict intermediate velocities which are then projected onto the space of divergence-free vector fields. By introducing more coupling between the diffusion--{;onvection step and the projection step we obtain a temporal discretiza­ tion that is second-order accurate. Our treatment of the diffusion-convection step uses a specialized higher order Godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high Reynolds number. The Godunov procedure is second-order accurate for smooth flow and remains stable for discontinuous initial data, even in the zero-viscosity limit. We approximate the projection directly using a Galerkin procedure that uses a local basis for discretely divergence-free vector fields. Numerical results are presented validating the convergence properties of the method. We also apply the method to doubly periodic shear-layers to assess the performance of the method on more difficult

1,163 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20228
2021151
2020209
2019182
2018207
2017213