707 research outputs found
An unstructured mesh control volume method for two-dimensional space fractional diffusion equations with variable coefficients on convex domains
In this paper, we propose a novel unstructured mesh control volume method to
deal with the space fractional derivative on arbitrarily shaped convex domains,
which to the best of our knowledge is a new contribution to the literature.
Firstly, we present the finite volume scheme for the two-dimensional space
fractional diffusion equation with variable coefficients and provide the full
implementation details for the case where the background interpolation mesh is
based on triangular elements. Secondly, we explore the property of the
stiffness matrix generated by the integral of space fractional derivative. We
find that the stiffness matrix is sparse and not regular. Therefore, we choose
a suitable sparse storage format for the stiffness matrix and develop a fast
iterative method to solve the linear system, which is more efficient than using
the Gaussian elimination method. Finally, we present several examples to verify
our method, in which we make a comparison of our method with the finite element
method for solving a Riesz space fractional diffusion equation on a circular
domain. The numerical results demonstrate that our method can reduce CPU time
significantly while retaining the same accuracy and approximation property as
the finite element method. The numerical results also illustrate that our
method is effective and reliable and can be applied to problems on arbitrarily
shaped convex domains.Comment: 18 pages, 5 figures, 9 table
Detailed analysis of the lattice Boltzmann method on unstructured grids
The lattice Boltzmann method has become a standard for efficiently solving
problems in fluid dynamics. While unstructured grids allow for a more efficient
geometrical representation of complex boundaries, the lattice Boltzmann methods
is often implemented using regular grids. Here we analyze two implementations
of the lattice Boltzmann method on unstructured grids, the standard forward
Euler method and the operator splitting method. We derive the evolution of the
macroscopic variables by means of the Chapman-Enskog expansion, and we prove
that it yields the Navier-Stokes equation and is first order accurate in terms
of the temporal discretization and second order in terms of the spatial
discretization. Relations between the kinetic viscosity and the integration
time step are derived for both the Euler method and the operator splitting
method. Finally we suggest an improved version of the bounce-back boundary
condition. We test our implementations in both standard benchmark geometries
and in the pore network of a real sample of a porous rock.Comment: 42 page
Non-local control in the conduction coefficients: well posedness and convergence to the local limit
We consider a problem of optimal distribution of conductivities in a system
governed by a non-local diffusion law. The problem stems from applications in
optimal design and more specifically topology optimization. We propose a novel
parametrization of non-local material properties. With this parametrization the
non-local diffusion law in the limit of vanishing non-local interaction
horizons converges to the famous and ubiquitously used generalized Laplacian
with SIMP (Solid Isotropic Material with Penalization) material model. The
optimal control problem for the limiting local model is typically ill-posed and
does not attain its infimum without additional regularization. Surprisingly,
its non-local counterpart attains its global minima in many practical
situations, as we demonstrate in this work. In spite of this qualitatively
different behaviour, we are able to partially characterize the relationship
between the non-local and the local optimal control problems. We also
complement our theoretical findings with numerical examples, which illustrate
the viability of our approach to optimal design practitioners
Grid generation for the solution of partial differential equations
A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given
Computational Engineering
This Workshop treated a variety of finite element methods and applications in computational engineering and expanded their mathematical foundation in engineering analysis. Among the 53 participants were mathematicians and engineers with focus on mixed and nonstandard finite element schemes and their applications
Unsteady incompressible Navier-Stokes simulations on deforming domains
Mesh generation has been an important topic of research for the past four decades, primarily because it is one of the critical elements in the numerical simulation of fluid flows. One of the main current issues in this regard is mesh generation and flow solution on domains with moving boundaries. In this research, a novel scheme has been proposed for mesh generation on domains with moving boundaries, with the location of boundary nodes known at any particular time. A new set of linearized equations is derived based on a full nonlinear elliptic grid generation system. The basic assumption in deriving these new equations is that each node experiences only a small amount of disturbance when the mesh moves from one time to the next. Comparison with grids generated by the full elliptic system shows that this new method can generate high quality grids with significantly less computational cost. Inherently, the flow on such a domain will be unsteady. The Navier-Stokes equations for unsteady 2D laminar incompressible flow are expressed in the primitive variables formulation. A SIMPLE-like scheme is applied to link the pressure and velocity fields and ensure conservation of mass is satisfied. The equations are discretized in a pure finite difference formulation and solved by implicitly marching in time. The flow solver is validated against results in the literature for flow through a channel with a moving indentation along one wall
Incompressible Lagrangian fluid flow with thermal coupling
In this monograph is presented a method for the solution of an incompressible viscous fluid flow with heat transfer and solidification usin a fully Lagrangian description on the motion. The originality of this method consists in assembling various concepts and techniques which appear naturally due to the Lagrangian formulation.Postprint (published version
Multi-dimensional higher resolution methods for flow in porous media.
Currently standard first order single-point upstream weighting methods are employed in reservoir simulation for integrating the essentially hyperbolic system components. These methods introduce both coordinate-line numerical diffusion (even in 1-D) and cross-wind diffusion into the solution that is grid and geometry dependent. These effects are particularly important when steep fronts and shocks are present and for cases where flow is across grid coordinate lines. In this thesis, families of novel edge-based and cell-based truly multidimensional upwind formulations that upwind in the direction of the wave paths in order to minimise crosswind diffusion are presented for hyperbolic conservation laws on structured and unstructured triangular and quadrilateral grids in two dimensions. Higher resolution as well as higher order multidimensional formulations are also developed for general structured and unstructured grids. The schemes are coupled with existing consistent and efficient continuous CVD (MPFA) Darcy flux approximations. They are formulated using an IMPES (Implicit in Pressure Explicit in Saturation) strategy for solving the coupled elliptic (pressure) and hyperbolic (saturation) system of equations governing the multi-phase multi-component flow in porous media. The new methods are compared with single point upstream weighting for two-phase and three-component two-phase flow problems. The tests arc conducted on both structured and unstructured grids and involve full-tensor coefficient velocity fields in homogeneous and heterogeneous domains. The comparisons demonstrate the benefits of multidimensional and higher order multidimensional schemes in terms of improved front resolution together with significant reduction in cross-wind diffusion
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