D3PLOT 22.1

"Simple Formula"

This method calculates results from a "simple" mathematical formula using built-in data components and standard maths operations.

All calculations are performed using floating point arithmetic and produce a single result, thus:

Scalar components are evaluated from a single formula
Vector components use three (independent) formulae: one for each of [x,y,z]
Tensor components use six formulae, one for each of[xx,yy,zz,xy,yz,zx]

Syntax rules for Simple Formulae

Formulae follow typical "computer programming" syntax, using commonly understood operators.

Operators supported:
*, /, +, -	Multiply, divide, plus and minus	* and / take precedence over + and -
** and ^	To the power of. Thus "x**2" is "x squared"	Note that "^" is Fortran syntax, not the C "exclusive OR" operator.
%	Modulus operator (x % y is remainder of x / y)	There is also a "mod" maths function.
( ... )	Brackets may be nested to any (reasonable) level	(...) take precedence over all other operators
Note: all arithmetic is floating point, thus both "5.0 / 2.0" and "5 / 2" will both deliver the answer 2.5

A formula must be a single line, up to 256 characters long.

Standard D3PLOT "built in" data components supported

Category	Acronym	Description	Acronym	Description
Nodal data	bx by bz cx cy cz dx dy dz dm vx vy vz vm ax ay az am	Basic X coordinate Basic Y coordinate Basic Z coordinate Current X coordinate Current Y coordinate Current Z coordinate X displacement Y displacement Z displacement Displacement magnitude X velocity Y velocity Z velocity Velocity magnitude X acceleration Y acceleration Z acceleration Acceleration magnitude	rdx rdy rdz rdm rvx rvy rvz rvm rax ray raz ram temp tbot tmid ttop tfx tfy tfz tfm dtdt	X rotation displacement Y rotation displacement Z rotation displacement Rotation displacement magnitude X rotation velocity Y rotation velocity Z rotation velocity Rotation velocity magnitude X rotation acceleration Y rotation acceleration Z rotation acceleration Rotation acceleration magnitude Temperature Nodal (shell) bottom surface temperature Nodal (shell) middle surface temperature Nodal (shell) top surface temperature X temperature flux Y temperature flux Z temperature flux Temperature magnitude dTemp / dTime
Solid and shell data	Stress and strain tensor-derived data is extracted in the global frame of reference unless the " l " (for "local") suffix is added to the component acronym, in which case it is extracted in the element local system. See below for how the Frame of Reference transformation applies to Simple Formula components.
Global tensor components	sxx syy szz sxy or syx syz or szy szx or sxz	XX stress YY stress ZZ stress XY stress ( symmetric ) YZ stress ( ditto ) ZX stress ( ditto)	exx eyy ezz exy or eyx eyz or ezy ezx or exz	XX strain YY strain ZZ strain XY strain ( symmetric ) YZ strain ( ditto ) ZX strain ( ditto )
Element Local tensor components	sxxl syyl szzl sxyl or syxl syzl or szyl szxl or sxzl	XX stress YY stress ZZ stress XY stress ( symmetric ) YZ stress ( ditto ) ZX stress ( ditto)	exxl eyyl ezzl exyl or eyxl eyzl or ezyl ezxl or exzl	XX strain YY strain ZZ strain XY strain ( symmetric ) YZ strain ( ditto ) ZX strain ( ditto )
Non-directional components derived from tensor data	svon smax smid smin sav sms evon emax emid emin eav ems	von Mises stress Max principal stress Mid principal stress Min principal stress Average stress (pressure) Max shear stress von Mises strain Max principal strain Mid principal strain Min principal strain Average strain (pressure) Max shear strain	epl erate pemag engmaj engmin engthk	Effective plastic strain Strain rate Plastic strain magnitude Engineering Major strain Engineering Minor strain Engineering Thickness strain
Non-directional components from Nastran OP2 file	sen senp send ken kenp kend	Strain energy Strain energy percentage Strain energy density Kinetic energy Kinetic energy percentage Kinetic energy density	enl enlp enld	Energy loss Energy loss percentage Energy loss density
Shell only data	rfx rfy rfxy rqx rqy rmx rmy rmxy s2max s2min e2max e2min eratio	Fx resultant force Fy resultant force Fxy resultant force Qx resultant force Qy resultant force Mx resultant moment My resultant moment Mxy resultant moment 2D (in plane) max princ stress 2D (in plane) min princ stress 2D (in plane) max princ strain 2D (in plane) min princ strain 2D (in plane) princ strain ratio	thk area eden hgen tstp mass madd	Shell Thickness Shell Area Internal energy density Hourglass energy Timestep Mass Added mass
Solid only data	erate	Strain rate	vol rvol	Element Volume Element Relative volume
Beam only data	bfx bfy bfz bfr bmxx bmyy bmzz brm bsxx bsyx or bsxy bszx or bszx bep beax bsax bpe1 bpe2 bry1 bry2 brz1 brz2 bmy1 bmy2 bmz1 bmz2 baen bie brxx	Axial force YY shear force ZZ shear force Resultant force XX torsional moment YY bending moment ZZ bending moment Resultant moment XX axial stress YX shear stress ( symmetric ) ZX shear stress ( ditto ) Plastic strain Axial strain Total axial strain Plastic energy at end 1 Plastic energy at end 2 Y rotation end 1 Y rotation end 2 Z rotation end 1 Z rotation end 2 Y moment end 1 Y moment end 2 Z moment end 1 Z moment end 2 Axial energy Internal energy Torsional rotation	bbed baed bied bsen bsenp bsend bken bkenp bkend benl benlp benld	Bending energy density Axial energy density Internal energy density Strain energy Strain energy percentage Strain energy density Kinetic energy Kinetic energy percentage Kinetic energy density Energy loss Energy loss percentage Energy loss density
"Extra" and ALE data	sox n shx n	Solid Extra component #n Shell Extra component #n	ammg n amss n adens adomf	Ale Multi-Matl group #n Mass of MM group #n Ale density Ale dominant fraction
Contact data	csn cst csx csy carea	Contact Normal Stress Contact Tangential stress Contact local X stress Contact local Y stress Contact segment area These are contact segment data averaged at nodes	cfgx cfgy cfgz cflx cfly cflz cfm	Contact global X force Contact global Y force Contact global Z force Contact local X force Contact local Y force Contact local Z force Contact force magnitude
LSDA (binout) data (Only available if both a ZTF file and a binout file have been read.)	sw_f sw_s sw_trsn sw_fail sw_time sp_f sp_e sp_m sp_r xsec_f_x xsec_f_y xsec_f_z xsec_m_x xsec_m_y xsec_m_z xsec_a	Spotweld axial force Spotweld shear force Spotweld torsion moment Spotweld failure Spotweld failure time Spring axial force Spring elongation Spring torsional moment Spring rotation Database X-sect X force ditto Y force ditto Z force ditto X moment ditto Y moment ditto Z moment Database X-sect area	sb_f sb_l sr_p rt_f rt_p spc_f_x spc_f_y spc_f_z spc_m_x spc_m_y spc_m_z	Seatbelt axial force Seatbelt length Slipring pull-through Retractor force Retractor pull-out SPC X force (at node) SPC Y ditto SPC Z ditto SPC X moment (at node) SPC Y ditto SPC Z ditto
User-defined components	unos n usss n ubms n	Nodal scalar component #n So/Sh scalar component #n Beam scalar component #n	Simple formulae may reference other user-defined components as well as the standard components above.
Material properties valid for Parts and part-based elements	dens ymod prat ystrs fstrn	Material density Young's modulus Poisson's ratio Yield stress Failure strain	These are calculated by PRIMER and written to the .ZTF file, so they will only be available if a ZTF file has been read. (A binout file is not required.) Not all properties are calculable for all material types, and -1.0 will be returned where values cannot be computed.

Applicability of the "Standard" data components in the table above

The data components above are all implicitly available for nodes.

Where the data is nodally-derived (eg current X coordinate cx ) then the value is used directly.
Where the data is element-derived (for example beam axial force bfx , or contact stress csn ) then the nodal value will be the average of all elements of the relevant type meeting at that node.

Elements may only use components of their "native" type, or nodally-derived data.

Where an element data component matches the element type (eg X stress sxx for shells, or YY bending moment bmyy for beams) then it is used directly.
Where an element references a nodally-derived data component (eg X displacement dx ) the value used will be the average of all nodes on that element.
Where an element references an element-derived data component of a different type (eg a shell element refers to beam axial force bfx ) then the result will be zero.

Applicability of programme settings: Integration point, frame of reference, etc.

Data is extracted "per integration point" for user-defined components, and it is not currently possible to specify data from some explicit integration point within a Simple Formula, meaning that you cannot use this method to assemble data from multiple integration points in an element.
The current global / local / cylindrical "frame of reference" (see REF_FRAME... Choosing the frame of reference ) does not apply when a Simple Formula value is calculated from raw tensor components (eg Sxx) for solids and shells. The unqualified acronym (Sxx, Syz, ...) always extracts global data, and the acronym with an "l" suffix (eg Sxxl, Sxzl) always extracts element local data.

However if you build a tensor Simple Formula (6 formulae) for solids or shells then the current "Frame of reference" is applied when it is used . In other words tensor Simple Formulae are created as "raw data", exactly like analysis data read from disk, and are subject to the same transformations when used. It is necessary to work this way in order to prevent "double transformations". (Scalar and vector user-defined components are not transformed.)

Maths functions supported

General functions	sqrt(x) log(x) (natural log) log10(x) (log base 10) exp(x) (e to the x)	mod(x,y) (modulo x / y) max(x1,x2) min(x1,x2) sign(x,y) (sign of y transferred to \|x\|)	abs(x) int(x) (truncate to integer) nint(x) (nearest integer) ceil(x) floor(x)
Trigonometric functions (use radians, not degrees)	sin(x) cos(x) tan(x)	sec(x) csc(x) cot(x)	asin(x) acos(x) atan(x) atan2(x,y)
Hyperbolic functions	sinh(x) cosh(x) tanh(x)		asinh(x) acosh(x) atanh(x)

Arithmetic types used

ALL calculations in simple formulae are processed using double precision floating point arithmetic. This includes constants, and means that the "integer arithmetic" conventions of languages such as Fortran or C, and their associated truncation, do not apply.

For example the expression 5 / 2 is converted to 5.0 / 2.0 , and so returns the result 2.5

Also the "integer" functions above ( int, nint, ceil , floor ) return integer values, but expressed as floating results.

For example nint(3.141592) evaluates to 3.0

Examples of simple formulae

temp - 273.15	Convert temperatures from Kelvin to Celsius
(sxx + syy + szz) / -3.0	Standard calculation for average stress
1 / sqrt(2) * sqrt((sxx-syy)2 + (syy-szz)2 + (szz-sxx)*2 + 6(sxy2 + syz2 + szx**2))	Standard calculation for von Mises stress
sxxexx + syyeyy + szzszz + 2.0sxyexy + 2.0syzeyz + 2.0szx*ezx	An estimate of strain energy density in a fully plastic section

Note that the "built in" data components described above can be referred to as simple acronyms, whereas maths functions require their arguments to be placed inside brackets.

Defining a scalar formula

In the case of a scalar component only a single formula is used.

In the example here the user is calculating pressure at nodes from stress components.

Defining a vector formula

In the case of a vector component three formulae are used, one for each of the [x,y,z] components. These are three wholly separate formulae which are evaluated independently.

In the example here the user is applying factors to displacements at nodes.

Note that each row must have a formula defined for the component to be valid.

Defining a tensor formula

In the case of a tensor component six formulae are used, one for each of the [xx,yy,zz,xy,yz,zx] components. These are six wholly separate formulae which are evaluated independently.

In the example here the user is extracting only the in-plane (2D) components of strain for an element.

As with vectors each row must have a formula defined for the component to be valid even if, as here, some row values are set explicitly to zero.

Further notes on Simple Formulae

Handling errors and missing data

If the expression being evaluated is grammatically legal, but not valid in a given context, then zero is returned. Typical examples are:

If a formula references a built-in data component that is not present in the model database.
If a mathematically illegal operation is attempted, such as divide by zero or the square root of a negative number.

Evaluation of element quantities at nodes

If an expression refers to an element data component, for example sxx , but is being evaluated for a node, then the relevant value is averaged from all legal elements at that node subject to the current averaging rules in force.

This can result in elements of more than one type being used: in this example component sxx , X stress, can be present in solids, shells and thick shells, so if multiple element types use the node then the value will contain contributions from all of them.

In general you should avoid contouring element quanties evaluated at nodes, since the effect is to smear out results and hence miss peak values at element centres. The result is identical to "low resolution" contouring of element data for scalar components.

Evaluation of nodal quantities at elements

If an expression refers to a nodal quantity, for example cx current X coordinate, but is being evaluated for an element, then the average value from all the nodes on that element is used.

Limitations of simple formulae

Simple formulae are powerful and very easy to use, but they do have limitations:

They only operate on the current node or element, at the current surface or layer if relevant.
They do not permit conditional expressions: you cannot use "if(...)" expression syntax

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