| Example code | oomph-lib features/conventions illustrated by the example code | Completeness of the documentation |
| Poisson problems |
The 1D Poisson equation
We (re-)solve the problem considered in the Quick Guide, this time using existing oomph-lib objects: The OneDMesh and finite elements from the QPoisson family. |
- General post-processing routines.
- General conventions:
- Use of public typedefs to specify function pointers.
- Element constructors should not have any arguments.
| complete |
The 2D Poisson equation.
We solve a 2D Poisson problem with Dirichlet boundary conditions and compare the results against an exact solution. |
- How to apply Dirichlet boundary conditions in complex meshes.
- How to use the
DocInfo object to label output files. - How does one change the linear solver in the Newton method?
- General conventions:
oomph-lib mesh objects are templated by the element type. How does one pre-compile mesh objects?
| complete
|
The 2D Poisson equation with flux boundary conditions (I)
Another 2D Poisson problem -- this time with Dirichlet and Neumann boundary conditions. |
- How to apply non-Dirichlet (flux) boundary conditions with
FaceElements.
- General conventions:
- What are broken virtual functions and why/when/where are they used?
| complete |
The 2D Poisson equation with flux boundary conditions (II)
An alternative solution for the previous problem, using multiple meshes. |
- How to use multiple meshes.
- General conventions:
- In problems with multiple sub-meshes,
Nodes retain the boundary numbers of the mesh in which they were created.
| complete |
| Poisson problems with adaptivity |
Adaptive solution of Poisson's equation in a fish-shaped domain
We solve a 2D Poisson equation in a nontrivial, fish-shaped domain and demonstrate oomph-lib's fully-automatic mesh adaptation routines. |
- How to perform automatic mesh adaptation.
oomph-lib's "black-box" adaptive Newton solver.The functions Problem::actions_before_adapt() and Problem::actions_after_adapt().
| complete |
The 2D Poisson equation revisited -- how to create a refineable mesh
We revisit an earlier example and demonstrate how easy it is to "upgrade" an existing mesh to a mesh that can be used with oomph-lib's automatic mesh adaptation routines. |
- "Upgrading" meshes to make them refineable.
- General conventions:
- Hanging nodes -- the functions
Node::position(...) and Node::value(...).
| complete |
Poisson's equation in a fish-shaped domain revisited -- mesh adaptation in deformable domains with curvilinear and/or moving boundaries.
We revisit an earlier example and demonstrate how to create refineable meshes for problems with curvilinear and/or moving domain boundaries. |
- How to create refineable meshes for problems with curvilinear and/or moving domain boundaries.
- General conventions:
- The
GeomObject, Domain and MacroElement objects. - The
Mesh::node_update() function. - It is good practice to store boundary coordinates for (
Boundary) Nodes that are located on curvilinear domain boundaries.
| complete |
Adaptive solution of Poisson's equation with flux boundary conditions.
We revisit an earlier example and demonstrate how to apply flux boundary conditions in problems with spatial adaptivity. |
- How to apply flux boundary conditions in problems with spatial adaptivity.
- General conventions:
- The
Mesh::flush_element_and_node_storage() function: "Emptying" a mesh without deleting its constituent nodes and elements (they might be shared with other meshes!).
| complete |
Adaptive solution of a 3D Poisson equations in a spherical domain
We demonstrate oomph-lib's octree-based 3D mesh adaptation routines. |
- Setting up and solving 3D problems isn't any harder than doing it in 2D.
- General conventions:
- The namespace
CommandLineArgs provides storage for the command line arguments to make them accessible throughout the code. - Plotting mesh boundaries.
| complete |
| The advection-diffusion equation |
The 2D advection diffusion equation with spatial adaptivity
We solve a 2D advection-diffusion equation and illustrate the characteristic features of solutions at large Peclet number. |
- The adaptive discretisation of the advection diffusion equation.
- How to specify the "wind" and the Peclet number for the advection diffusion equation.
- How to document the progress of
oomph-lib's "black box" adaptive Newton solver.
| complete |
2D advection diffusion equation with Neumann (flux) boundary conditions.
We solve a 2D advection-diffusion equation with flux boundary conditions. |
- How to specify Neumann (flux) boundary conditions for the advection diffusion equation.
| complete |
The 2D advection diffusion equation revisited: Petrov-Galerkin methods and SUPG stabilisation.
We demonstrate how to implement a stabilised Petrov-Galerkin discretisation of the advection diffusion equation. |
- Petrov-Galerkin discretisation of the advection-diffusion equation.
- General conventions:
- The role of shape, basis and test functions.
- The element building blocks: Geometric elements, equation classes and specific elements.
| Driver code and pretty pictures |
| The unsteady heat equation and an introduction to time-stepping |
The 2D unsteady heat equation
We solve the 2D unsteady heat equation and demonstrate oomph-lib's time-stepping procedures for parabolic problems. |
- Solving time-dependent problems with
Problem::unsteady_newton_solve(...) - The functions
Problem::actions_before_implicit_timestep() and Problem::actions_before_implicit_timestep(). - The BDF timesteppers and how to set up initial conditions for parabolic problems.
- Initialising the "previous" nodal positions for elements that are based on an ALE formulation.
- General conventions:
- Providing a default
Steady timestepper for all Mesh constructors. - Steady and unsteady versions of functions -- position and interpretation of the (discrete) time index.
| complete |
The 2D unsteady heat equation with restarts
We demonstrate oomph-lib's dump/restart capabilities which allow time-dependent simulations to be restarted. |
- The functions
Problem::dump(...) and Problem::read(...). - How to customise the generic dump/restart functions.
| complete |
The 2D unsteady heat equation with adaptive timestepping
We demonstrate oomph-lib's adaptive timestepping capabilities. |
- How to enable temporal adaptivity.
- The function
Problem::adaptive_unsteady_newton_solve(). - The function
Problem::global_temporal_error_norm(). - How to choose the target for the global temporal error norm.
| complete |
Spatially adaptive solution of the 2D unsteady heat equation with Neumann (flux) boundary conditions.
We solve a 2D unsteady heat equation in a non-trivial domain with flux boundary conditions and compare the computated results against the exact solution. |
- Spatial adaptvity in time-dependent problems.
- Choosing the maximum number of spatial adaptations per timestep.
- Using
Problem::set_initial_condition() to assign initial conditions ensures that the initial conditions are re-assigned when mesh adaptations are performed while the first timestep is computed. - The functions
Problem::dump(...) and Problem::read(...) can handle adaptive meshes.
| complete |
Spatially adaptive solution of the 2D unsteady heat equation in a moving domain with Neumann (flux) boundary conditions.
We demonstrate the spatially adaptive solution of a 2D unsteady heat equation in a nontrivial moving domain. |
- The ALE form of the unsteady heat equation and its implementation in
oomph-lib's unsteady heat elements. - The role of the positional
TimeStepper and the importance of assigning history values for the nodal positions.
- General conventions:
- The function
Mesh::node_update() automatically updates the nodal positions in response to the deformation/motion of time-dependent GeomObjects that define the Domain and Mesh boundaries.
| complete |
Spatially and temporally adaptive solution of the 2D unsteady heat equation in a moving domain with flux boundary conditions.
We demonstrate the use of combined spatial and temporal adaptivity for the solution of a 2D unsteady heat equation in a nontrivial moving domain. |
- Adaptive timestepping combined with spatial adaptivity.
| complete |
| The linear wave equation |
The 2D linear wave equation.
We solve a 2D wave equation and demonstrate oomph-lib's time-stepping procedures for hyperbolic problems. |
- Timestepping for hyperbolic problems: The Newmark scheme.
- How to set up initial conditions for hyperbolic problems.
- Default settings for the linear wave equation elements.
- How to apply Neumann (flux) boundary conditions for the linear wave equation.
| complete |
The spatially-adaptive solution of the 2D linear wave equation
We demonstrate that the spatially-adaptive solution of the 2D wave equation is difficult (in fact, not very sensible) because any spurious waves that are generated during mesh adaptation are never damped out. Refineable linear wave elements should therefore only be used to "manually" refine meshes before the start of the simulation. |
- How to use refineable linear wave elements
| driver code only |
| The Young Laplace equation |
The solution of the Young-Laplace equation.
We solve the Young Laplace equation that governs the shape of static air-liquid interfaces. |
- Theory and implementation.
- The use of spines for the representation of complicated air-liquid interfaces.
- Natural boundary conditions along free contact lines.
- How to use displacement control to compute (past) limit points in the load-displacement characteristics.
| complete |
Contact-angle boundary conditions for the Young-Laplace equation
We demonstrate how to apply contact angle-boundary conditions for the Young-Laplace equation. |
- Theory and implementation of contact angle-boundary conditions.
- How to generate initial guesses for the solution
- Limitations of the current approach and suggestions for improvement
- An inherent difficulty in problems with zero contact angles.
| complete |
| The Navier-Stokes equations |
The 2D Navier-Stokes equations: Driven cavity flow
Probably the most-solved problem in computational fluid dynamics: Steady driven cavity flow. We illustrate the problem's discretisation with Taylor-Hood and Crouzeix-Raviart elements. |
- Discretising the steady Navier-Stokes equations: The governing equations and their implementation in the stress-divergence and simplified forms.
- Non-dimensional parameters and their default values.
- The pressure representation in Taylor-Hood and Crouzeix-Raviart elements.
- Pinning a pressure value in problems with Dirichlet boundary conditions for the velocity on all boundaries.
| complete |
The 2D Navier-Stokes equations: Adaptive solution of the 2D driven cavity problem
We employ oomph-lib's mesh adaptation routines to refine the mesh in the neighbourhood of the pressure singularities. |
- Treatment of pressure degrees of freedom in Navier-Stokes simulations with adaptive mesh refinement -- pinning "redundant" pressure degrees of freedom.
| complete |
The 2D Navier-Stokes equations: Driven cavity flow in a quarter-circle domain with mesh adaptation
We re-solve the driven-cavity problem in a different domain, demonstrate how to apply body forces and show how to switch between the stress-divergence and simplified forms of the Navier-Stokes equations. |
- Adapting the driven cavity problem to different domains.
- How to apply body forces in the Navier-Stokes equations.
- How to switch between the stress-divergence and the simplified form of the incompressible Navier-Stokes equations.
| complete |
Adaptive simulation of 3D finite Reynolds number entry flow into a circular pipe
We solve the classical problem of entry flow into a 3D tube. |
- Adaptivity for 3D Navier-Stokes problems
- How to determine the numbering scheme for mesh boundaries.
- How to adjust parameters that control the behaviour of
oomph-lib's Newton solver. - The natural (traction-free) boundary conditions for the Navier-Stokes equations.
| complete |
A variant of Rayleigh's oscillating plate problem: The unsteady 2D Navier-Stokes equations with periodic boundary conditions
We solve a variant of the classical Rayleigh plate problem to demonstrate the use of periodic boundary conditions and time-stepping for the Navier-Stokes equations. |
- Timestepping for the Navier-Stokes equations.
- How to apply periodic boundary conditions.
- General conventions:
- Periodic boundary conditions should be applied in the
Mesh constructor.
| complete |
Another variant of Rayleigh's oscillating plate problem: The unsteady 2D Navier-Stokes equations with periodic boundary conditions, driven by an applied traction.
We demonstrate how to apply traction boundary conditions for the Navier-Stokes equations. |
- How to apply traction boundary conditions for the Navier-Stokes equations.
| complete |
2D finite-Reynolds-number-flow driven by an oscillating ellipse
We study the 2D finite-Reynolds number flow contained inside an oscillating elliptical ring and compare the computed results against an exact solution (an unsteady stagnation point flow). |
- Solving the Navier-Stokes equations in a moving domain.
- How to apply no-slip boundary conditions on moving walls, using the function
FSI_functions::apply_no_slip_on_moving_wall(...)
| complete |
2D finite-Reynolds-number-flow in a 2D channel with a moving wall
This is a "warm-up" problem for the classical fluid-structure interaction problem of flow in a 2D collapsible channel. Here we compute the flow through a 2D channel in which part of one wall is replaced by a moving "membrane" whose motion is prescribed. |
- The adaptive solution of the Navier-Stokes equations in a moving domain with traction boundary conditions.
| complete |
2D finite-Reynolds-number-flow in a 2D channel with a moving wall revisited: Algebraic Node updates.
We re-visit the problem studied in the previous example and demonstrate an alternative node-update procedure, based on oomph-lib's AlgebraicNode, AlgebraicElement and AlgebraicMesh classes. Algebraic node updates will turn out to be essential for the efficient implementation of fluid-structure interaction problems. |
- How to customise the node-update, using
oomph-lib's AlgebraicNode, AlgebraicElement and AlgebraicMesh classes. - Existing
AlgebraicMeshes are easy to use: Simply "upgrade" the required element (of type ELEMENT, say) in the templated wrapper class AlgebraicElement<ELEMENT>.
| complete |
2D finite-Reynolds-number-flow in a 2D channel that is partially obstructed by an oscillating leaflet
This is a "warm-up" problem for the corresponding fluid-structure interaction problem where the leaflet deforms in response to the fluid traction. Here we consider the case where the motion of the leaflet is prescribed. |
- Another example illustrating the use of algebraic and
MacroElement/Domain-based node update techniques.
| complete |
Flow past a cylinder with a waving flag
This is a "warm-up" problem for Turek & Hron's FSI benchmark problem where the flag deforms in response to the fluid traction. Here we consider the case where the motion of the flag is prescribed. |
- Another example illustrating the use of algebraic and
MacroElement/Domain-based node update techniques.
| complete |
Unstructured meshes for fluids problems
This is a "warm-up" problem for another tutorial in which we demonstrate the use of unstructured meshes for FSI problems. |
- How to use xfig/triangle-generated, unstructured meshes for flow problems.
| complete |
Unstructured meshes for 3D fluids problems
This is a "warm-up" problem for another tutorial in which we demonstrate the use of unstructured 3D meshes for FSI problems. |
- How to use tetgen-generated, unstructured meshes for 3D flow problems.
- Avoiding "locking" with the
split_corner_elements flag.
| complete |
Steady finite-Reynolds-number flow through an iliac bifurcation
We show how to simulate physiological flow problems, using the Vascular Modeling Toolkit (VMTK). This is a "warm-up" problem for another tutorial in which we consider the corresponding FSI problems in which the vessel wall is elastic. |
- How to use
ImposeParallelOutflowElements to enforce parallel in- and outflow from cross-sections that are not aligned with any coordinate planes.
| complete |
Adaptive simulation of flow at finite Reynolds number in a curved circular pipe
We solve the classical problem of flow into a 3D curved tube. |
- Another example of adaptivity for 3D Navier-Stokes problems
| driver code and pretty pictures
|
| The axisymmetric Navier-Stokes equations |
Spin-up of a viscous fluid -- the spatially adaptive solution of the unsteady axisymmetric Navier-Stokes equations.
A classical fluid mechanics problem: Spin-up of a viscous fluid. A key feature of the flow is the development of thin Ekman (boundary) layers during the early stages of the spin-up. We demonstrate how the use of spatial adaptivity helps to resolve these layers. At large times, the flow field approaches a rigid-body rotation -- this poses a subtle problem for the spatial adaptivity as its default behaviour would cause strong spatially uniform refinement. |
- The axisymmetric Navier-Stokes equations
- How to prescribe a constant reference value for the normalisation of the error in spatially-adaptive computations in which the solution approaches a "trivial" solution.
| driver code and pretty pictures
|
| The free-surface Navier-Stokes equations |
The Bretherton problem: An air finger propagates into a 2D fluid-filled channel.
A classical fluid mechanics problem: We study the propagation of an inviscid (air) finger into a 2D fluid-filled channel and compare our results against those from Bretherton's theoretical analysis. |
| driver code and pretty pictures |
Free-surface relaxation oscillations of a viscous fluid layer.
We study the oscillations of perturbed fluid layer. |
| driver code and pretty pictures |
Relaxation oscillations of an interface between two viscous fluids.
We study the oscillations of two-layer fluid system. |
| driver code only |
A static free surface bounding a layer of viscous fluid.
A hydrostatics problem: Compute the static free surface that bounds a layer of viscous fluid -- harder than you might think! |
| driver code only |
A static interface between two viscous fluids.
A hydrostatics problem: Compute the static interface between two viscous fluids -- harder than you might think! |
| driver code only |
A steadily-rotating cylinder below a free surface in a finite box.
Compute the free surface position and fluid velocity and pressure fields about a fixed, steadily-rotating cylinder immersed in a viscous fluid in a finite box. Uses a pseudo-elastic remesh strategy and spatial adaptivity in a non-trivial free surface problem. |
| driver code only
|
| Solid mechanics problems |
Solid mechanics: Theory and implementation
In this document we discuss the theoretical background and the practical implementation of oomph-lib's solid mechanics capabilities. |
- Theory:
- Solid mechanics problems -- Lagrangian coordinates
- The geometry
- Equilibrium and the Principle of Virtual Displacements
- Constitutive Equations for Purely Elastic Behaviour
- Non-dimensionalisation
- 2D problems: Plane strain.
- Isotropic growth.
- Specialisation to a cartesian basis and finite element discretisation
- Implementation:
- The
SolidNode class - The
SolidFiniteElement class - The
SolidMesh class - The
SolidTractionElement class
- Timestepping and the generation of initial conditions for solid mechanics problems
| complete |
Bending of a cantilever beam
We study a classical solid mechanics problem: the bending of a cantilever beam subject to a uniform pressure loading on its upper face and/or gravity. We compare the results for zero-gravity against the (approximate) analytical St. Venant solution for the stress field. |
- How to formulate solid mechanics problems.
- How to choose a constitutive equation.
- How to apply traction boundary conditions, using
SolidTractionElements.
| complete |
Axisymmetric compression of a circular disk
We study the axisymmetric compression of a circular, elastic disk, loaded by an external traction. The results are compared against the predictions from small-displacement elasticity. |
- How to upgrade an existing mesh to
SolidMesh. - Why it is necessary to use "undeformed MacroElements" to ensure that the numerical results converge to the correct solution under mesh refinement if the domain has curvilinear boundaries.
- How to switch between different constitutive equations
- how to incorporate isotropic growth into the model.
| complete |
Compressible and incompressible behaviour
We discuss various issues related to (in)compressible material behaviour and illustrate various solution techniques in a simple test-problem: The compression of a square block of (compressible or incompressible) material by a gravitational body force. The results are compared against the predictions from small-displacement elasticity. |
- Different formulations of the constitutive laws for compressible, near-incompressible and incompressible behaviour.
- How to combine the various constitutive laws with the displacement and pressure/displacement formulations of the principle of virtual displacements.
- The default setting: Incompressibility is not enforced automatically. It must be requested explicitly.
| complete |
Axisymmetric oscillations of a circular disk
We study the free axisymmetric oscillations of a circular, elastic disk and compare the eigenfrequencies and modes against the predictions from small-displacement elasticity. |
- How to assign initial conditions for unsteady solid mechanics problems.
| complete |
Deformation of a solid by a prescribed boundary motion
We study the large deformations of a 2D elastic domain, driven by the prescribed deformation of its boundary. The boundary motion is imposed by Lagrange multipliers. This technique is important for the solution of fluid-structure interaction problems in which the deformation of the fluid mesh is controlled by (pseudo-)elasticity. |
- How to impose displacement boundary conditions in solid mechanics problems by Lagrange multipliers.
| complete |
Large-amplitude bending of an asymmetric 3D cantilever beam made of incompressible material.
We study the deformation of an asymmetric 3D canvilever beam made of incompressible material. |
- How to enforce incompressible behaviour.
- How to solve 3D solid mechanics problems with spatial adaptivity.
| complete |
Unstructured meshes for 2D and 3D solid mechanics problems.
We demonstrate how to use unstructured meshes to solve 2D and 3D solid mechanics problems. This tutorial acts as a "warm-up" problem for the solution of unstructured FSI problems. |
- How to solve 2D and 3D solid mechanics problems on unstructured meshes.
- How to identify domain boundaries in xfig-generated unstructured meshes.
| complete |
Unstructured meshes for 3D solid mechanics problems.
We demonstrate how to use unstructured meshes to solve 3D solid mechanics problems. This is a "warm-up" problem for another tutorial in which we demonstrate the use of unstructured 3D meshes for FSI problems. |
- How to solve 3D solid mechanics problems on unstructured meshes.
| complete |
Inflation of a blood vessel
We show how to simulate physiological solid mechanics problems, using the Vascular Modeling Toolkit (VMTK). This is a "warm-up" problem for another tutorial in which we consider the corresponding FSI problems in which the vessel conveys (and is loaded by) a viscous fluid. |
- How to solve physiological solid mechanics problems.
| complete |
Large-amplitude shock waves in a circular disk
We study the propagation of shock waves in an elastic 2D circular disk. |
- How to employ spatial adaptivity in time-dependent solid mechanics problems.
| driver code only |
Large shearing deformations of a hyper-elastic, incompressible block of material
We solve a classical problem in large-displacement elasticity and comprare against Green and Zerna's exact solution. |
| driver code only |
| Beam structures |
The deformation of a pre-stressed elastic beam, loaded by a pressure load
We study the lateral deformation of a pre-stressed elastic beam, using oomph-lib's geometrically non-linear Kirchhoff-Love-type HermiteBeamElement and compare the results against an (approximate) analytical solution. |
- How to specify the undeformed reference shape for the Kirchhoff-Love-type beam elements.
- How to apply boundary conditions and loads for the
HermiteBeamElement. - General conventions:
- How to change control parameters for the Newton solver.
| complete |
Large-displacement post-buckling of a pressure-loaded, thin-walled elastic ring
We compute the post-buckling deformation of a thin-walled elastic ring, subjected to a pressure load and compare the results against results from the literature. |
- How to use
oomph-lib's DisplacementControlElement to apply displacement control in solid mechanics problems.
- General conventions:
- What should be stored in a
GeneralisedElement's "external" Data? - What should be stored in a
Problem's "global" Data?
| complete |
Large-amplitude oscillations of a thin-walled elastic ring.
We compute the free, large-amplitude oscillations of a thin-walled elastic ring and demonstrate that Newmark's method is energy conserving. |
- How to assign initial conditions for beam structures.
- How to use the dump/restart function for
HermiteBeamElements. - Demonstrate that
Newmark timesteppers can be used with variable timesteps. - How to retrieve solutions at previous timesteps in computations with
Newmark timesteppers. - Changing the default non-dimensionalsation for time.
- The non-dimensionalisation of the kinetic and potential (strain) energies of
HermiteBeamElements.
| complete |
Small-amplitude oscillations of a thin-walled elastic ring.
We compute the free, small-amplitude oscillations of a thin-walled elastic ring, demonstrate that Newmark's method is energy conserving, and compare the oscillation frequencies and mode shapes against analytical predictions. |
- How to assign initial conditions for beam structures.
- General conventions:
| driver code only |
| Shell structures |
Large-displacement post-buckling of a clamped, circular cylindrical shell.
We simulate the post-buckling deformation of a pressure-loaded, clamped, thin-walled elastic shell. |
- How to specify the undeformed reference configuration with
HermiteShellElement structures. - Using displacement control for
HermiteShellElements. - General conventions:
| driver code only |
| (Fluid-structure) interaction problems |
Warm-up problem for free-boundary problems: How to parametrise unknown boundaries.
We demonstrate how to "upgrade" a GeomObject to a GeneralisedElement so that it can be used to parameterise an unknown domain boundary. |
- How to use multiple inheritance to combine
GeomObjects and GeneralisedElements. - How to "upgrade" a
GeomObject to a GeneralisedElement so that it can be used to parameterise an unknown domain boundary. - What is a
GeomObject's geometric Data? - What is a
GeneralisedElement's external and internal Data?
| complete |
A toy interaction problem: The solution of a 2D Poisson equation coupled to the deformation of the domain boundary
We show how to use MacroElementNodeUpdateElements and MacroElementNodeUpdateMeshes to implement sparse node update operations in free-boundary problems. We demonstrate their use in a simple free-boundary problem: The solution of Poisson's equation, coupled to an equation that determines the position of the domain boundary. |
- How to use
MacroElementNodeUpdateElements and MacroElementNodeUpdateMeshes to implement sparse node update operations in free-boundary problems. - Basic free-boundary problems: Making the domain boundary dependent on the solution in the domain.
- General conventions:
MacroElementNodeUpdateElements and MacroElementNodeUpdateMeshes.
| complete |
A classical fluid-structure interaction problem: Finite Reynolds number flow in a 2D channel with an elastic wall.
We demonstrate the solution of this classical fluid-structure interaction problem and demonstrate how easy it is to combine the two single-physics problems (the deformation of an elastic beam under pressure loading and the flow in a 2D channel with a moving wall) to a fully-coupled fluid-structure interaction problem. |
- The
FSIFluidElements and FSIWallElement base classes. - Representing a discretised beam/shell structure as a "compound"
GeomObject: The MeshAsGeomObject class. - Using the function
FSI_functions::setup_fluid_load_info_for_solid_elements(...) to set up the fluid-structure interaction. - The pros (convenient!) and cons (slow!) of the
MacroElement/Domain - based node-update procedures in fluid-structure interaction problems.
| complete |
Finite Reynolds number flow in a 2D channel with an elastic wall revisited: Sparse algebraic node updates
We revisit the problem of flow in a collapsible channel to demonstrate that the sparse algebraic node update procedures first discussed in an earlier non-FSI example lead to a much more efficient code. |
- How to "sparsify" the node update with algebraic node update procedures.
- The
GeomObject::locate_zeta(...) function and its default implementation.
| complete |
Finite Reynolds number flow in a 2D channel with an elastic wall revisited again: Spatial adaptivity in fluid-structure interaction problems.
We revisit the problem of flow in a collapsible channel yet again to demonstrate the use of spatial adaptivity in fluid-structure interaction problems. |
- How to use spatial adaptativity in fluid-structure interaction problems.
- The
Steady<NSTEPS> timestepper: How to assign positional history values for newly created nodes. - Updating the node-update data in refineable
AlgebraicMeshes.
| complete |
Segregated solvers for fluid-structure-interaction problems: Revisiting the flow in a 2D collapsible channel
We revisit the problem of flow in a collapsible channel once more to demonstrate the use of segregated solvers in fluid-structure interaction problems. |
- The base SegregatableFSIProblem class
- How to construct and solve a segregated problem from a (standard) "monolithic" problem
| complete |
Preconditioning monolithic solvers for fluid-structure-interaction problems: Revisiting the flow in a 2D collapsible channel yet again
We revisit the problem of flow in a collapsible channel yet again to demonstrate the use of oomph-lib's FSI preconditioner for the monolithic solution of fluid-structure interaction problems. |
- How to use
oomph-lib's FSIPreconditioner
| complete |
Flow past a flexible leaflet
We study the flow in a 2D channel that is partially obstructed by an elastic leaflet. |
- FSI problems with fully immersed beam structures (i.e. beams that are subjected to the fuid traction on both faces).
- Another application of
oomph-lib's FSIPreconditioner.
| complete |
Turek & Hron's FSI benchmark: Flow past an elastic flag attached to a cylinder
We demonstrate how to discretise and solve this benchmark problem with oomph-lib. |
- FSI problems with "proper" 2D solids (rather than beam or shell structures)
- FSI problems with wall inertia.
- Another application of
oomph-lib's FSIPreconditioner.
- Note: There is a separate tutorial that discusses the parallelisation of this code and shows to modify it to allow spatial adaptativity in the fluid and solid meshes.
| complete |
Using unstructured meshes for FSI problems.
We demonstrate how to use xfig/triangle-generated unstructured meshes for fluid-structure interaction problems. |
- How to use xfig/triangle-generated unstructured meshes for fluid-structure interaction problems.
- The automatic generation of boundary coordinates for xfig/triangle-generated, unstructured meshes. How it works and what can go wrong...
| complete |
Using unstructured meshes for 3D FSI problems.
We demonstrate how to use tetgen-generated unstructured meshes for 3D fluid-structure interaction problems. |
- How to use tetgen-generated unstructured meshes for 3D fluid-structure interaction problems.
- The automatic generation of boundary coordinates for tetgen-generated, unstructured meshes. How it works and what can go wrong...
| complete |
Finite-Reynolds-number flow through an elastic iliac bifurcation
We show how to simulate physiological fluid-structure interaction problems, using the Vascular Modeling Toolkit (VMTK). |
- How to attach multiple
FaceElements to the same node -- distinguishing different Lagrange multipliers. - How to solve physiological fluid-structure interaction problems.
| complete |
A simple fluid-structure interaction problem: Finite Reynolds number flow, driven by an oscillating ring.
This is a very simple fluid-structure interaction problem: We study the finite-Reynolds number internal flow generated by an oscillating ring. The wall motion only has a single degree of freedom: The ring's average radius, which needs to be adjusted to conserve mass. The nodal positions in the fluid domain is updated by MacroElements. [This is a warm-up problem for the full fluid structure interaction problem discussed in the next example]. We compare the predictions for the flow field against asymptotic results. |
- Fluid-structure interaction.
- General conventions:
| driver code only |
A simple fluid-structure interaction problem re-visited: Finite Reynolds number flow, driven by an oscillating ring -- this time with algebraic updates for the nodal positions.
We re-visit the simple fluid-structure interaction problem considered in the earlier example.This time we perform the update of the nodal positions with AlgebraicElements. |
- Fluid-structure interaction.
- General conventions:
| driver code only |
A real fluid-structure interaction problem: Finite Reynolds number flow in an oscillating elastic ring.
Our first "real" fluid-structure interaction problem: We study the finite-Reynolds number internal flow generated by the motion of an oscillating elastic ring and compare the results against asymptotic predictions. |
- Fluid-structure interaction.
- General conventions:
| driver code only |
| Multi-physics problems |
Simple multi-physics problem: How to combine exisiting single-physics elements into new multi-physics elements.
We demonstrate how to "combine" a CrouzeixRaviartElements and QAdvectionDiffusionElements into a single BuoyantCrouzeixRaviartElement that solves the Navier--Stokes equations under the Boussinesq approximation coupled to an energy equation. |
- How to use multiple inheritance to combine two single-physics elements.
- How to write single-physics elements that can be combined into multi-physics elements.
- How to use the
Problem::steady_newton_solve(...) function to find steady solutions of unsteady problems.
| complete |
Refineable multi-physics problem: How to combine exisiting refineable single-physics elements into new refineable multi-physics elements.
We demonstrate how to "combine" a RefineableCrouzeixRaviartElements and RefineableQAdvectionDiffusionElements into a single RefineableBuoyantCrouzeixRaviartElement that solves the Navier--Stokes equations under the Boussinesq approximation coupled to an energy equation. |
- How to use multiple inheritance to combine two refineable single-physics elements.
- How to choose the "Z2 flux" for multi-physics elements that are both derived from the
ElementWithZ2ErrorEstimator class.
| complete |
Solving multi-field problems with multi-domain discretisations.
We demonstrate an alternative approach to the solution of multi-field problems, in which the governing PDEs are discretised in separate meshes and interact via "external elements". |
- How to discretise multi-field problems with multi-domain approaches.
- The
Multi_domain_functions namespace.
| complete |
Thermoelasticity: How to combine single-physics elements with solid mechanics elements.
We demonstrate how to "combine" a QUnsteadyHeatElement and QPVDElement into a single QThermalPVDElement that solves the equations governing elastic deformations coupled to uniform thermal expansion. The geometric coupling back to the heat equation is completely hidden. |
- How to use multiple inheritance to combine a single-phyics element and a solid element
| driver code and pretty pictures |
