Intrinsic modal solution¤

Overall solution process¤

The tensor structure of the main components in the solution process is illustrated in the figure below in the sequential order they are computed, together with the asymptotic time and space complexities. The discretization of the problem comprises $N_n$ number of condensed nodes, $N_m$ modes used in the reduced order model and $N_t$ time steps in the solution (if the problem is static, $N_t$ represents a ramping load stepping scheme). The intrinsic modes, $\Phi, \Psi \in \mathbb{R}^{N_m \times 6 \times N_n}$ are computed from the eigenvalue solution and the coordinates $\pmb{x}_a \in \mathbb{R}^{3 \times N_n}$ of the active nodes. The nonlinear couplings, $\pmb{\Gamma} \in \mathbb{R}^{N_m \times N_m \times N_m}$ are calculated next, from which the is assembled and solved to yield the solution states $\pmb q \in \mathbb{R}^{N_t \times N_s}$ with $N_s$ the number of states in the system that is proportional to the number of modes. Local velocities, internal forces and strain fields $\pmb{x}_{1,2,3} \in \mathbb{R}^{N_t \times 6 \times N_n}$ are computed as a product of the corresponding intrinsic modes and states, and their integration leads to the position tensor, $r_a$ with similar structure. In some cases, such as when gravity forces are included, the evolution of the rotational matrix, $\pmb R_{ab}$ , needs to be solved for too.

Intrinsic modal solution

For links to the codebase, see the following:

Modes.
Nonlinear couplings.
Solution of equations:
- Orchestrator to build the Systems
- Numerical solvers inside Sollibs: Using Diffrax or bespoke solvers in JAX.
- Right-hand-side (RHS) of the system of equations implemented in pure functions to comply with JAX functional programming approach.

System based solutions¤

An internal function identifier (Sol name) is build such that unique functions are mapped to the input settings in the system of equations, thereby avoiding the use of if-conditionals, 0-only additional vectors inside the solvers (running iteratively). The options are as follows:

Type	Target	Gravity	BC1 [prime=2]	ModalAero [prime=3]	SteadyAero [prime=5]	UnsteadyAero [prime=7]	Point loads [prime=11]	q0 approx	Rigid-body	Nonlinearities	residualised
1 static	Level	False: "g"	Clamped	None	None	None	None	via q2	1-quaternion+strains	All -> ""	None -> ""
2 Dynamic	TRIM1	True: "G"	Free	Rogers	qalpha	gust	follower	via q1	All-quaternions	Linear sys -> "l"	True -> "r"
3 Stability	TRIM2		Prescribed	Loewner	qx (control)	controls	dead			Linear sys+disp -> "L"
4 Control										only gamma1 -> "g1"

And some of the implemented RHS solutions (see for instance equations):

Sol name		label	Imp
10G1	Structural static under Gravity	[1,0,G]	Y
10g11	Structural static with follower point forces	[1,0,g,0,0,0,0,1]	Y
10g121	Structural static with dead point forces	[1,0,g,0,0,0,0,2]	Y
10g1331	Structural static with follower+dead forces	[1,0,g,0,0,0,0,3]	N
10g15	Manoeuvre under qalpha	[1,0,g,0,1,1]	Y
10G15	Manoeuvre under qalpha and Gravity	[1,0,G,0,1,1]	N
10g75	Manoeuvre under qalpha and controls	[1,0,g,0,1,2]	N
10G75	Manoeuvre under qalpha+controls+Gravity	[1,0,G,0,1,2]	N
20g1	Clamped Structural dynamics, free vibrations	[2,0,g]	Y
20G2	Free Structural dynamic with gravity forces	[2,0,G,1]	Y
20g2	Free Structural dynamic	[2,0,g,1]	Y
20g11	Structural dynamic follower point forces	[2,0,g,0,0,0,0,1]	Y
20g121	Structural dynamic dead point forces	[2,0,g,0,0,0,0,2]	Y
20g22	Free Structural dynamic follower point forces	[2,0,g,1,0,0,0,1]	Y
20g242	Free Structural dynamic dead point forces	[2,0,g,1,0,0,0,2]	Y
11G6	Static trimmed State (elevator-qalpha,	[1,1,G,1,1]	Y
	no gravity updating)		Y
12G2	Static trimmed State (elevator-qalpha,	[1,2,G,1]	Y
	gravity updating)
21G150	Dynamic trimmed State	[2,1,G,1,1,2]	N
20g21	Gust response	[2,0,g,0,1,0,1]	Y
20g273	Gust response, q0 obtained via integrator q1	[2,0,g,0,1,0,1,0,1]	Y
20g105	Gust response with steady qalpha	[2,0,g,0,1,1,1]	N
20g42	Gust response Free-flight	[2,0,g,1,1,0,1]	Y
20G42	Gust response Free-flight and gravity (X error)	[2,0,G,1,1,0,1]	N
20G1050	Gust response Free-flight, gravity, controls	[2,0,G,1,1,2,1]	N

Sol name is based on the input options obtained as the multiplication of prime numbers, which can be obtained and inspected as:

import feniax.intrinsic.functions as functions
label = functions.label_generator([2,0,'g',0,1,0,1,0,1])
print(label)

Recovery of deformations¤

Analytical solutions to the are obtained when the strain is assumed constant between nodes, using a piecewise constant integration. If a component in the load-path is discretized in $n$+1 points, strain and curvatures are defined in the mid-points of the spatial discretization (n in total). $\gamma_n$ and $\kappa_n$ are constant within the segment $s_{n-1} \leq s \leq s_n$ , and the position and rotation matrix after integration are $\begin{equation} \begin{split} \textbf{R}_{ab}(s) &= \textbf{R}_{ab}(s_{n-1})\pmb{\mathcal{H}}^0(\pmb\kappa,s) \\ \pmb{r}(s) &= \pmb{r}(s_{n-1}) + \pmb{R}_{ab}(s_{n-1})\pmb{\mathcal{H}}^1(\pmb\kappa, s)\left(\pmb{e}_1+\pmb{\gamma}_n\right) \end{split} \end{equation}$ with the operators $\pmb{\mathcal{H}}^0(\pmb\kappa, s)$ and $\pmb{\mathcal{H}}^1(\pmb\kappa, s)$ obtained from integration of the exponential function, as in \cite{PALACIOS2010}. Note that when position and rotations are recovered from strain integration, there is still one point that is either clamped or needs to be tracked from integration of its local velocity.

Algorithm