****************************** Convective Lagrangian Velocity ****************************** :: Convective Lagrangian Velocity = {model_name} {float_list} [L/t] ----------------------- **Description / Usage** ----------------------- In solid mechanics, when the deformation of the mesh is Lagrangian, i.e., motion of the solid can be described by a mapping from the stress-free state (undeformed state) to the deformed state, it is often desirable to prescribe a convective velocity of the stress-free state that can lead to inertial forces through deformation (see Technical Discussion below). This required card allows for the specification of solid-body translation or rotation of the stress-free state, and results in an inertial term on the otherwise quasi static solid momentum equation. Definitions of the input parameters are as follows: +-------------+---------------------------------------------------------------------------------------+ |{model_name} |Name of the prescribed velocity model. This parameter can have one of the following | | |values: **NONE**, **CONSTANT**, or **ROTATIONAL**. | +-------------+---------------------------------------------------------------------------------------+ |{float_list} |One or more floating point numbers ( through ) whose values are | | |determined by the selection for {model_name}. These are identified in the discussion of| | |each model below. Note that not all models employ a {float_list}. | +-------------+---------------------------------------------------------------------------------------+ Thus, +--------------------------------------------------+---------------------------------------------------------+ |**NONE** |the stress-free state is assumed to be unmoving. No | | |floating point input values are required with this model.| +--------------------------------------------------+---------------------------------------------------------+ |**CONSTANT** |the stress-free state is one of solid-body translation, | | |viz. it moves uniformly with a velocity specified by | | |three orthogonal components: | | | | | | * - X-component of velocity | | | * - Y-component of velocity | | | * - Z-component of velocity (for 3-D) | +--------------------------------------------------+---------------------------------------------------------+ |**ROTATIONAL** |the stress-free state is one of solid-body rotation at a | | |specified rotation rate. | | | | | | * - Rotation rate, in radians/sec. | | | * - X-position of axis of rotation (must be | | | constant in 3D). | | | * - Y-position of axis of rotation (must be | | | constant in 3D, viz. the axis must be perpendicular to| | | both the X and Y axes, viz. the axis must be the Z | | | axis. | | | * - Set to zero. Parameter is not used for now.| | | | | |Note that this model is applicable in 2-D and certain 3-D| | |problems in which the rotation axis is the Z-axis. To | | |generalize this model to three-dimensions, the proper | | |input will require a point and a direction of the | | |rotation axis. In two-dimensions, the axis of rotation | | |is the Z-direction. | +--------------------------------------------------+---------------------------------------------------------+ ------------ **Examples** ------------ The following is a sample input card: :: Convective Lagrangian Velocity = ROTATIONAL 25.0 1. 1. 0. This card is associated with a material file, and hence a material that is of *LAGRANGIAN* or *TOTAL_ALE* type (see *Mesh Motion* card). That material’s stressfree state, as specified by this model, will rotate about an axis that is located at [1.0, 1.0, 0] at 25 radians/sec (assuming seconds are the time scale of the problem). ------------------------- **Technical Discussion** ------------------------- This capability is often used when problems require a force or a boundary condition to be applied to a solid material that is moving relative to the source, or the desired frame of reference. Such constraints arise mainly in fluid-structure interaction problems where one solid material is moving relative to another, with a fluid material in between, e.g. deformable blade or knife metering/pushing liquid over a flat or round substrate. These models have also been used in porous-material translation relative to a drying source (see references below). Specification of any model but **NONE** on this card produces the left-hand-side term in the equation for quasi static equilibrium: .. figure:: /figures/358_goma_physics.png :align: center :width: 90% :math:`\sigma` is the Cauchy stress tensor of the solid material, and f is the body force per unit volume. The first term is a result of the specified advection of the stress-free state. :math:`v_m^0`, which depends solely on the user-prescribed velocity and the current state of deformation, is by definition .. figure:: /figures/359_goma_physics.png :align: center :width: 90% where :math:`F_m` is the material deformation gradient tensor (computed somewhat differently depending on the formulation, as described in the references below), and :math:`v_sfs` is the stress-free state velocity field specified by this card. -------------- **References** -------------- SAND96-2149: Drying in Deformable Partially-Saturated Porous Media: Sol-Gel Coatings, Cairncross, R. A., P. R. Schunk, K. S. Chen, S. S. Prakash, J. Samuel, A. J. Hurd and C. Brinker (September 1996) GT-005.3: THE NEW TOTAL-ARBITRARY-LAGRANGIAN-EULERIAN (TALE) CAPABILITY and its applicability to coating with/on deformable media, August 6, 1999, P. R. Schunk SAND2000-0807: TALE: An Arbitrary Lagrangian-Eulerian Approach to Fluid- Structure Interaction Problems, P. R. Schunk (May 2000) .. TODO - Line 97 and 106 are photos need to be replaced with the actual equations.