In this article we further investigate the solution of linear second order elliptic boundary value problems by distributed Lagrange multipliers based fictitious domain methods. The following issues ...are addressed: (i) Derivation of the fictitious domain formulations. (ii) Finite element approximation. (iii) Iterative solution of the resulting finite dimensional problems (of the saddle-point type) by preconditioned conjugate gradient and Lanczos algorithms.
In this article we discuss a methodology that allows the direct numerical simulation of incompressible viscous fluid flow past moving rigid bodies. The simulation methods rest essentially on the ...combination of:
(a) Lagrange-multiplier-based fictitious domain methods which allow the fluid flow computations to be done in a fixed flow region.
(b) Finite element approximations of the Navier–Stokes equations occurring in the global model.
(c) Time discretizations by operator splitting schemes in order to treat optimally the various operators present in the model.
The above methodology is particularly well suited to the direct numerical simulation of particulate flow, such as the flow of mixtures of rigid solid particles and incompressible viscous fluids, possibly non-Newtonian. We conclude this article with the presentation of the results of various numerical experiments, including the simulation of store separation for rigid airfoils and of sedimentation and fluidization phenomena in two and three dimensions.
This computational study shows, for the first time, a clear transition to two-dimensional Hopf bifurcation for laminar incompressible flows in symmetric plane expansion channels. Due to the ...well-known extreme sensitivity of this study on computational mesh, the critical Reynolds numbers for both the known symmetry-breaking (pitchfork) bifurcation and Hopf bifurcation were investigated for several layers of mesh refinement. It is found that under-refined meshes lead to an overestimation of the critical Reynolds number for the symmetry breaking and an underestimation of the critical Reynolds number for the Hopf bifurcation.
In this article, we discuss the numerical solution of the Dirichlet problem for the Monge–Ampère equation in two dimensions. The solution of closely related problems is also discussed; these include ...a family of Pucci’s equations, the equation prescribing the harmonic mean of the eigenvalues of the Hessian of a smooth function of two variables, and a minimization problem from nonlinear elasticity, where the cost functional involves the determinant of the gradient of vector-valued functions. To solve the Monge–Ampère equation we consider two methods. The first one “reduces”the Monge–Ampère equation to a saddle-point problem for a well-chosen augmented Lagrangian; to solve this saddle-point problem we advocate an Uzawa–Douglas–Rachford algorithm. The second method combines nonlinear least-squares and operator-splitting. This second method being simpler to implements, we apply variants of it to the solution of the other problems. For the space discretization we use mixed finite element approximations, closely related to methods already used for the solution of linear and nonlinear bi-harmonic problems; through these approximations the solution of the above problems is, essentially, reduced to the solution of discrete Poisson problems. The methods discussed in this article are validated by the results of numerical experiments.
The main goal of this article is to discuss a numerical method for finding the best constant in a Sobolev type inequality considered by C. Sundberg, and originating from Operator Theory. To simplify ...the investigation, we reduce the original problem to a parameterized family of simpler problems, which are constrained optimization problems from Calculus of Variations. To decouple the various differential operators and nonlinearities occurring in these constrained optimization problems, we introduce an appropriate augmented Lagrangian functional, whose saddle-points provide the solutions we are looking for. To compute these saddle-points, we use an Uzawa–Douglas–Rachford algorithm, which, combined with a finite difference approximation, leads to numerical results suggesting that the best constant is about five times smaller than the constant provided by an analytical investigation.
The main goal of this article is to investigate the capability of an operator-splitting/finite elements based methodology at handling accurately incompressible viscous flow at large Reynolds number ...(Re) in regions with corners and curved boundaries. To achieve this goal the authors have selected a wall-driven flow in a semi-circular cavity. On the basis of the numerical experiments reported in this article it seems that the method under investigation has no difficulty at capturing the formation of primary, secondary and tertiary vortices as Re increases; it has also the capability of identifying a Hopf bifurcation phenomenon taking place around Re=6600.
A Lagrange-multiplier-based fictitious-domain method (DLM) for the direct numerical simulation of rigid particulate flows in a Newtonian fluid was presented previously. An important feature of this ...finite element based method is that the flow in the particle domain is constrained to be a rigid body motion by using a well-chosen field of Lagrange multipliers. The constraint of rigid body motion is represented by
u=
U+
ω×
r
;
u
being the velocity of the fluid at a point in the particle domain;
U
and
ω are the translational and angular velocities of the particle, respectively; and
r
is the position vector of the point with respect to the center of mass of the particle. The fluid–particle motion is treated implicitly using a combined weak formulation in which the mutual forces cancel. This formulation together with the above equation of constraint gives an algorithm that requires extra conditions on the space of the distributed Lagrange multipliers when the density of the fluid and the particles match. In view of the above issue a new formulation of the DLM for particulate flow is presented in this paper. In this approach the deformation rate tensor within the particle domain is constrained to be zero at points in the fluid occupied by rigid solids. This formulation shows that the state of stress inside a rigid body depends on the velocity field similar to pressure in an incompressible fluid. The new formulation is implemented by modifying the DLM code for two-dimensional particulate flows developed by others. The code is verified by comparing results with other simulations and experiments.
We study memory states of a circuit consisting of a small inductively coupled Josephson junction array and introduce basic (write, read, and reset) memory operations logics of the circuit. The ...presented memory operation paradigm is fundamentally different from conventional single quantum flux operation logics. We calculate stability diagrams of the zero-voltage states and outline memory states of the circuit. We also calculate access times and access energies for basic memory operations.
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