5. Shell Elements

Tip: Try to use continuum elements where possible for better accuracy and reduced complexity for model setup/troubleshooting. Shell elements with dielectric should be a last resort.

Due to the disparity in dimensions between the width and thickness of the membrane (>100:1), it becomes computationally prohibitive to use continuum elements to represent the thickness of the membrane, with issues such as the following arising:

• Need an aspect ratio of 4:1 or less for accuracy
• Membrane needs to be at least 3 or 4 elements through the thickness
• Membrane is usually in the electric window and electrostatic solver, therefore more elements means longer computational time
• This leads to a very small timestep in the calculation for a large number of elements in the electric window

Thankfully, the solution to this problem is to simply replace the elements that make up the membrane with Shell elements

• Assumes very thin compared to width
• Analytically considers thickness effects
• Element width determines the timestep
• Only one 'node' thick

There are some specialist requirements that must be taken into account when using Shell elements, some of which have already been touched upon but they will be expanded here:

• Shells have rotational Degrees of Freedom (continuum do not) therefore when connecting a row of Shell elements to continuum elements, the Shells must extend 1 element into the continuum elements to prevent excessive flexibility

• They do not include electrical effects, they are mechanical only. Therefore 'Fake' continuum elements must be used to provide the dielectric behavior.

  • 'Fake' elements have zero mass, zero stiffness
  • Include correct dielectric properties for electric field calculation
  • To prevent a mechanical discontinuity in the model these 'Fake' elements are only 1 element thick and are then bonded together to prevent this
• They must be placed at appropriate height for bending (i.e. they must have room to bend)
• Shells should be placed along the Bending Axis if possible
• There must be the correct separation of electrodes to represent the electric field
• Intervening materials must have the correct dielectric properties
• Intervening materials must also have the correct thickness

Finally, Shell elements require a both a thickness and a number of layers to be specified in their definition. General rule of thumb is 20 layers and 20 elements along the membrane width. More elements may be required along the width if higher order modes are required.

Plotting Shells

Plotting Shell elements and Shell stresses/strains can be controlled by the grph set, grph plot and grph actv commands respectively. When plotting Shell arrays, the following table is very useful.

The set-up of shell elements follows the structure of the following code:

/* Shell element definitions
shel
    /* Since shell material properties are input as Young's modulus and Poisson ratio
    /* put in logic to compute these from wavespeed information

    symb cp = 8296.          /* longitudinal wave speed .
    symb cs = 5196.          /* shear wave speed   
    symb rho = 2328.         /* mass density
    symb b = $cp * $cp * $rho
    symb g = $cs * $cs * $rho
    symb young = $g * ( 3. * $b - 4. * $g ) / ( $b - $g )
    symb v = ( $b - 2. * $g ) / 2. / ( $b - $g )

    /* Define silicon nitride material for use in shell definition
    /* Note: Anisotropic properties can be used (TYPE ORTH)
    prop sin  $rho  $young $v  

    /* Define damping for shell elements
    avis sin matr * * * *

    /* Define a layer definition composed of silicon nitride layers total thickness is $ThkMem
    layr  mbrn $ThkMem 20 sin  

    /* Place shell elements in the model to represent the membrane
    elem  mbrn $i1 $i3 $j4 $j4  

    prnt                     /* print information about shell elements
    end