18,625 research outputs found
Numerical approximation of phase field based shape and topology optimization for fluids
We consider the problem of finding optimal shapes of fluid domains. The fluid
obeys the Navier--Stokes equations. Inside a holdall container we use a phase
field approach using diffuse interfaces to describe the domain of free flow. We
formulate a corresponding optimization problem where flow outside the fluid
domain is penalized. The resulting formulation of the shape optimization
problem is shown to be well-posed, hence there exists a minimizer, and first
order optimality conditions are derived.
For the numerical realization we introduce a mass conserving gradient flow
and obtain a Cahn--Hilliard type system, which is integrated numerically using
the finite element method. An adaptive concept using reliable, residual based
error estimation is exploited for the resolution of the spatial mesh.
The overall concept is numerically investigated and comparison values are
provided
Least-squares finite elements for distributed optimal control problems
We provide a framework for the numerical approximation of distributed optimal
control problems, based on least-squares finite element methods. Our proposed
method simultaneously solves the state and adjoint equations and is
-- stable for any choice of conforming discretization spaces. A
reliable and efficient a posteriori error estimator is derived for problems
where box constraints are imposed on the control. It can be localized and
therefore used to steer an adaptive algorithm. For unconstrained optimal
control problems, i.e., the set of controls being a Hilbert space, we obtain a
coercive least-squares method and, in particular, quasi-optimality for any
choice of discrete approximation space. For constrained problems we derive and
analyze a variational inequality where the PDE part is tackled by least-squares
finite element methods. We show that the abstract framework can be applied to a
wide range of problems, including scalar second-order PDEs, the Stokes problem,
and parabolic problems on space-time domains. Numerical examples for some
selected problems are presented
H‐adaptive finite element solution of high Rayleigh number thermally driven cavity problem
An h‐adaptive finite element code for solving coupled Navier‐Stokes and energy equations is used to solve the thermally driven cavity problem. The buoyancy forces are represented using the Boussinesq approximation. The problem is characterised by very thin boundary layers at high values of Rayleigh number (>106). However, steady state solutions are achievable with adequate discretisation. This is where the auto‐adaptive finite element method provides a powerful means of achieving optimal solutions without having to pre‐define a mesh, which may be either inadequate or too expensive. Steady state and transient results are given for six different Rayleigh numbers in the range 103 to 108 for a Prandtl number of 0.71. The use of h‐adaptivity, based on a posteriori error estimation, is found to ensure a very accurate problem solution at a reasonable computational cost
An adaptive Uzawa FEM for Stokes: Convergence without the Inf-Sup
We introduce and study an adaptive finite element method for the Stokes system based on an Uzawa outer iteration to update the pressure and an elliptic adaptive inner iteration for velocity. We show linear convergence in terms of the outer iteration counter for the pairs of spaces consisting of continuous finite elements of degree 푘 for velocity whereas for pressure the elements can be either discontinuous of degree 푘-1 or continuous of degree 푘-1 and 푘. The popular Taylor-Hood family is the sole example of stable elements included in the theory, which in turn relies on the stability of the continuous problem and thus makes no use of the discrete inf-sup condition. We discuss the realization and complexity of the elliptic adaptive inner solver, and provide consistent computational evidence that the resulting meshes are quasi-optimal
An Improved Continuous Sensitivity Equation Method for Optimal Shape Design in Mixed Convection
International audienceThis paper presents an optimal shape design methodology for mixed convection problems. The Navier-Stokes equations and the Continuous Sensitivity Equations (CSE) are solved using an adaptive finite-element method to obtain flow and sensitivity fields. A new procedure is presented to extract accurate values of the flow derivatives at the boundary, appearing in the CSE boundary conditions for shape parameters. Flow and sensitivity information are then employed to calculate the value and gradient of a design objective function. A BFGS optimization algorithm is used to find optimal shape parameter values. The proposed approach is first verified on a problem with a closed form solution, obtained by the method of manufactured solutions. The method is then applied to determine the optimal shape of a model cooling system
Convergence and optimality of the adaptive nonconforming linear element method for the Stokes problem
In this paper, we analyze the convergence and optimality of a standard
adaptive nonconforming linear element method for the Stokes problem. After
establishing a special quasi--orthogonality property for both the velocity and
the pressure in this saddle point problem, we introduce a new prolongation
operator to carry through the discrete reliability analysis for the error
estimator. We then use a specially defined interpolation operator to prove
that, up to oscillation, the error can be bounded by the approximation error
within a properly defined nonlinear approximate class. Finally, by introducing
a new parameter-dependent error estimator, we prove the convergence and
optimality estimates
Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity
Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and devise the hybridizable discontinuous Galerkin (HDG) method. Exact geometry described by non-uniform rational B-splines (NURBS) is integrated into HDG using the framework of the NURBS-enhanced finite element method (NEFEM). Moreover, optimal convergence and superconvergence properties of HDG-Voigt formulation in presence of symmetric second-order tensors are exploited to construct inexpensive error indicators and drive degree adaptive procedures. Applications involving the numerical simulation of problems in electrostatics, linear elasticity and incompressible viscous flows are presented. Moreover, this is done for both high-order HDG approximations and the lowest-order framework of face-centered finite volumes (FCFV).Peer ReviewedPostprint (author's final draft
POD model order reduction with space-adapted snapshots for incompressible flows
We consider model order reduction based on proper orthogonal decomposition
(POD) for unsteady incompressible Navier-Stokes problems, assuming that the
snapshots are given by spatially adapted finite element solutions. We propose
two approaches of deriving stable POD-Galerkin reduced-order models for this
context. In the first approach, the pressure term and the continuity equation
are eliminated by imposing a weak incompressibility constraint with respect to
a pressure reference space. In the second approach, we derive an inf-sup stable
velocity-pressure reduced-order model by enriching the velocity reduced space
with supremizers computed on a velocity reference space. For problems with
inhomogeneous Dirichlet conditions, we show how suitable lifting functions can
be obtained from standard adaptive finite element computations. We provide a
numerical comparison of the considered methods for a regularized lid-driven
cavity problem
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