# Pavel

(Difference between revisions)
 Revision as of 14:21, 16 July 2013 (view source) (→Introduction)← Older edit Revision as of 14:22, 16 July 2013 (view source)Newer edit → Line 1: Line 1: − === Introduction === − The numerical solution of convection-diffusion transport problems arises in many important applications in science and engineering. These problems occur in many applications such as in the transport of air and ground water pollutants, oil reservoir flow, in the modeling of semiconductors, and so forth. − − Theory − − Under the assumption of incompressibility, the governing equations are given by − − [[Image:placa.jpg|thumb|right|300px|Problem description]] − $− \rho C \frac{\partial T}{\partial t} + \rho C v \cdot \nabla T + \nabla − \cdot q = 0 −$ − (1) − − − − $− q = -\kappa \nabla \cdot T −$ − (2) − − In the context of mass difussion within a fluid, (1) is is the mass − conservation equation and (2) is a contitutive law proposed by Fourier. The − notation is standard: $\rho$ is the density, $C$ the heat capacity, $\kappa$ − the thermal conductivity, $T$ is the temperature, $v$ is the velocity field − and $q$ is the diffusive flux per unit fluid density. − − − Remark: systen can be decoupled since we can plug (2) into (1) and solve the − scalar equation − − $− \rho C \frac{\partial T}{\partial t} + \rho Cv \cdot \nabla T - − \nabla \cdot (\kappa \nabla \cdot T) = 0 −$ − (3) − − − Problem statement − − Let us consider the transport by convection and diffusion in an open set − $\Omega$ $\subset \Re^d$(d=2 or 3) \ with piecewise smooth boundary $\Gamma$, − such that $\Gamma = \Gamma_d \cup \Gamma_N$. The unit outward normal vector to − $\Gamma$ is denoted $n$. The convection-diffusion initial-boundary value − problem can be stated as follows: given a divergence-free velocity field a, − the diffusion tensor $\kappa$ and adequate initial and boundary conditions, find $T$ : − $\bar{\Omega} \times \left[ 0, T \left] \rightarrow \Re \right. \right.$ such − that − − $− \rho C \frac{\partial T}{\partial t} + \rho Cv \cdot \nabla T - − \nabla \cdot (\kappa \nabla T) = 0 in \Omega \times (0, T) −$ − (4) − − − $− T(x, 0) = T_0 (x) on \Omega −$ − − $− T = T_D on \Gamma_D \times (0, T) −$ − − $− k (\nabla T) \cdot n = q on \Gamma_N \times (0, T) −$ − − − − Space discretization method − Multiplying Eq.(4) by a test function $W$ and intehrating on the whole domain − $\Omega$ the equation reads − − − $\int_\Omega \rho C \frac{\partial T}{\partial t} W d V + \int_\Omega \rho Cv \cdot − \nabla T W d V = \int_\Omega (\nabla \cdot (k \nabla T)) W d V −$ − (5) − − Integratin by parts the right term of Eq.(5) leads to − − $\int_\Omega \rho C \frac{\partial T}{\partial t} W d V + \int_\Omega v \cdot \nabla T W d − V = - \int_\Omega k \nabla T \cdot \nabla W d V$ (6) − − Finite element discretization − − The temperature is discretized in the standard finite element method manner as − $− T = \sum N_i T_i −$(7) − where $N_i$ are the nodal shape functions. Substituting the finite element − approximation (7) into the variational equation () and choosing a Galerling formulation ($W_i=N_i$) leads to the following equation: − − $\rho C M \frac{\partial T}{\partial t} + \rho C S T − = - \kappa L T$ (7) − − Time discretization method − Consider a first-order BDF (that is, the Euler implicit scheme) − − − $− \frac{\partial T}{\partial t}=\frac{(T^{n + 1} - T^n)}{\Delta t} −$ − or a second-order BDF − − − $− \frac{\partial T}{\partial t}= \frac{1}{2} \frac{(3 \times T^{n + 1} - 4 \times T^n + T^{n-1})}{\Delta t} −$ − − == Computational Structural Mechanics module == − − === Introduction === − − Examples showing the class of problems that the code can solve (2-4 examples) − − Description of the underlying theory and schematic list of the problems this application can solve. − − − The Computational Structural Mechanics module (CSM) is.... − − === Application Structure === − − ==== Analysis Type ==== − The available solutions strategies are: − * '''Static''' − * '''Dynamic''' − * '''Relaxed dynamic''' − − With this module you can solve both '''linear''' and '''non linear''' problems. − In case of non linear problems several methods are available: − * '''Newton-Raphson''' − * '''Newton Raphson with line search''' − * '''Arch lenght''' − − Different solvers are availables (LINK TO SOLVER SECTION!!!!) − − − ==== Elements ==== − * '''Frame Elements''': − ** '''Euler-Bernoulli beam'''  short explanation − ** '''Crisfield truss''' short explanation − − * '''2D elements''' − ** '''Linear triangular element''': − − * '''Shell elements''': − ** '''Isotropic shell''': (change the name with the usual one!!!!) − ** '''Ansotropic shell''':  (change the name with the usual one!!!!) − ** '''EBST shell''':  (change the name with the usual one!!!!) − − * '''Membrane element''': − − * '''Solid elements''': − ** '''Linear tetrahedral element''': − − {| class="wikitable" width="100%" style="text-align:left; background:#d0d9dd; border:0px solid #e1eaee; font-size:100%; -moz-border-radius-topleft:0px; -moz-border-radius-bottomleft:0px; padding:0px 0px 0px 0px;" valign="top" − !Dimension − !Element Type − !Kratos name − !Geometry − !Nonlinearity − !Material Type − |-style="background:#F1FAFF;" − | rowspan="1" | 1D − | Frame − | [[LinearBeamElement]] − | Line − | − | Isotropic − |-style="background:#F1FAFF;" − | rowspan="1" | 1D − | Truss − | [[CrisfieldTrussElement]] − | Line − | Large Displacement − | Isotropic − |-style="background:#F1FAFF;" − | rowspan="1" | 2D − | Solid − | [[TotalLagrangian]] − | 2D Geometries − | Large Displacement − | Isotropic − |-style="background:#F1FAFF;" − | rowspan="4"| 3D − | Solid − | [[TotalLagrangian]] − | 3D Geometries − | Large Displacement − | Isotropic − |-style="background:#F1FAFF;" − | Shell − | [[ShellIsotropic]] − | 3D Triangle − | Large Displacement − | Isotropic − |-style="background:#F1FAFF;" − | Shell − | [[ShellAnisotropic]] − | 3D Triangle − | Large Displacement − | Orthotropic − |} − − ==== Boundary Conditions ==== − − Boundary conditions can be set fixing displacements and rotations degrees of freedom. − − ==== Loads ==== − − * Self weight − * Punctual force − * Moment − * Face pressure (sign convenction!!!!) − * Distributed load − − ==== Constitutive laws ==== − − The following constitutive laws are available: − − * '''Linear elastic''': − * ... − − ==== HPC ==== − The code can be run in shared or distributed memory: − * OpenMP: − * MPI: − − ==== Problem parameters ==== − − ... − − ==== Others relevand aspects ==== − − ... − − === Benchmarking === − − Here validation and verification examples should be inserted − − === Tutorials === − − === Contact people === − − === Akcnowledgements === == Convection Diffusion module == == Convection Diffusion module ==

## Convection Diffusion module

### Introduction

The numerical solution of convection-diffusion transport problems arises in many important applications in science and engineering. These problems occur in many applications such as in the transport of air and ground water pollutants, oil reservoir flow, in the modeling of semiconductors, and so forth. This paper describes the Convection Diffusion Applications for solving this equation.

Theory

Under the assumption of incompressibility, the governing equations are given by $\rho C \frac{\partial T}{\partial t} + \rho C v \cdot \nabla T + \nabla \cdot q = 0$ (1) $q = -\kappa \nabla \cdot T$ (2)

In the context of mass difussion within a fluid, (1) is is the mass conservation equation and (2) is a contitutive law proposed by Fourier. The notation is standard: ρ is the density, C the heat capacity, κ the thermal conductivity, T is the temperature, v is the velocity field and q is the diffusive flux per unit fluid density.

Remark: systen can be decoupled since we can plug (2) into (1) and solve the scalar equation $\rho C \frac{\partial T}{\partial t} + \rho Cv \cdot \nabla T - \nabla \cdot (\kappa \nabla \cdot T) = 0$ (3)

Problem statement

Let us consider the transport by convection and diffusion in an open set Ω $\subset \Re^d$(d=2 or 3) \ with piecewise smooth boundary Γ, such that $\Gamma = \Gamma_d \cup \Gamma_N$. The unit outward normal vector to Γ is denoted n. The convection-diffusion initial-boundary value problem can be stated as follows: given a divergence-free velocity field a, the diffusion tensor κ and adequate initial and boundary conditions, find T : $\bar{\Omega} \times \left[ 0, T \left] \rightarrow \Re \right. \right.$ such that $\rho C \frac{\partial T}{\partial t} + \rho Cv \cdot \nabla T - \nabla \cdot (\kappa \nabla T) = 0 in \Omega \times (0, T)$ (4)

T(x,0) = T0(x)onΩ $T = T_D on \Gamma_D \times (0, T)$ $k (\nabla T) \cdot n = q on \Gamma_N \times (0, T)$

Space discretization method Multiplying Eq.(4) by a test function W and intehrating on the whole domain Ω the equation reads $\int_\Omega \rho C \frac{\partial T}{\partial t} W d V + \int_\Omega \rho Cv \cdot \nabla T W d V = \int_\Omega (\nabla \cdot (k \nabla T)) W d V$ (5)

Integratin by parts the right term of Eq.(5) leads to $\int_\Omega \rho C \frac{\partial T}{\partial t} W d V + \int_\Omega v \cdot \nabla T W d V = - \int_\Omega k \nabla T \cdot \nabla W d V$ (6)

Finite element discretization

The temperature is discretized in the standard finite element method manner as $T = \sum N_i T_i$(7) where Ni are the nodal shape functions. Substituting the finite element approximation (7) into the variational equation () and choosing a Galerling formulation (Wi = Ni) leads to the following equation: $\rho C M \frac{\partial T}{\partial t} + \rho C S T = - \kappa L T$ (7)

Time discretization method Consider a first-order BDF (that is, the Euler implicit scheme) $\frac{\partial T}{\partial t}=\frac{(T^{n + 1} - T^n)}{\Delta t}$

or a second-order BDF $\frac{\partial T}{\partial t}= \frac{1}{2} \frac{(3 \times T^{n + 1} - 4 \times T^n + T^{n-1})}{\Delta t}$

### Structure

#### Analysis type

The available solution strategy is:

   Dynamic


With this module you can solve both linear and non linear problems.

#### Kinematical approaches

Eulerian and Lagrangian approach are available in order to solve the equation.

#### Elements

Linear triangular elements in 2D and linear tetrahedra elements in 3D. Both elements are stabilized with OSS.

ConvDiff2D

ConvDiff3D

#### Boundary conditions

Dirichlet boundary condition: $T = T_D on \Gamma_D \times (0, T)$

Neumann boundary conditions: $k (\nabla T) \cdot n = q on \Gamma_N \times (0, T)$

#### Initial conditions

Initial condition in temperature can be set.

#### HPC

The code can be run in shared or distributed memory:

• OpenMP:
• MPI:

#### Problem parameters

The parameters involved in this problem are:

ρ : Density

C :heat capacity

κ: thermal conductivity

v : velocity field

q: diffusive flux per unit fluid density.

T a: ambient temperature.

σ: Stefen Boltzmann constant

e: emissivity

h: convection coefficient