MORFE.jl

MORFE.jl implements the Direct Parametrisation of Invariant Manifolds (DPIM) — a high-order, spectral-submanifold-based method for nonlinear model order reduction of large finite element models.

Why MORFE.jl?

Large finite-element models of nonlinear structures routinely reach $n \sim 10^5$ degrees of freedom. Direct time-integration or frequency-response continuation at this scale is prohibitively expensive, while classical linear modal truncation misses essential nonlinear phenomena (hardening, softening, internal resonances).

MORFE.jl resolves this by computing a polynomial parametrisation of a low-dimensional Spectral Submanifold (SSM) that:

• captures exact nonlinear dynamics on the manifold — not just a linear approximation;
• reduces the system to $m \sim 2$–$10$ coordinates regardless of FE model size;
• handles harmonic and quasi-periodic external forcing via autonomous external variables;
• is FEM-agnostic — works with any backend that provides $K$, $C$, $M$ and nonlinear force vectors.

Installation

MORFE.jl is not yet registered in the Julia General Registry. Install directly from GitHub:

using Pkg
Pkg.add(url="https://github.com/MORFEproject/MORFE.jl.git")

Quick Start

using MORFE
using LinearAlgebra, StaticArrays

# 1. Define system matrices  M ẍ + C ẋ + K x = F_nl(x, ẋ)
B0 = [2.0 -1.0; -1.0 2.0]   # stiffness
B1 = [0.01  0.0;  0.0 0.01] # damping
B2 = [1.0   0.0;  0.0 1.0]  # mass

# 2. Nonlinear term: cubic stiffness  −β x³
using MORFE.FullOrderModel: MultilinearMap, NDOrderModel
term_cubic = MultilinearMap(
    (res, x1, x2, x3) -> (@. res += -1.0 * x1 * x2 * x3),
    (3, 0),
)
model = NDOrderModel((B0, B1, B2), (term_cubic,))

# 3. Solve the generalised eigenproblem
using MORFE.FullOrderModel: linear_first_order_matrices
A, B = linear_first_order_matrices(model)
result = eigen(A, B)

# 4. Select master modes and build multiindex set
using MORFE.Multiindices: all_multiindices_up_to
ROM = 2;  NVAR = ROM;  p = 5
mset = all_multiindices_up_to(NVAR, p; min_degree = 1)

# 5. Build resonance set and solve the parametrisation
using MORFE.Resonance: resonance_set_from_graph_style
using MORFE.CohomologicalEquations: solve_cohomological_problem
# (see demo/ or notebooks/ for the complete pipeline)

How It Works

         Full-order FEM model (n ~ 10⁵ DOF)
                     │
             Spectral decomposition
          (generalised eigenproblem)
                     │
            Select m master modes
        (least-damped conjugate pairs)
                     │
         Build polynomial multiindex set
            𝒜 ⊂ ℕᵐ⁺ᴺᵉˣᵗ, degree ≤ p
                     │
         ┌───────────────────────────┐
         │  Cohomological solve loop │
         │  for each α ∈ 𝒜 (GrLex)  │
         │    s = ⟨λ, α⟩            │
         │    [A−sB]W[α] = N[α]     │  ← invariance equation
         │    (augmented at resonance)│
         └───────────────────────────┘
                     │
        Parametrisation W : ℂᵐ → ℂⁿ  (SSM embedding)
        Reduced dynamics R : ℂᵐ → ℂᵐ (flow on SSM)
                     │
         Post-process: FRC, backbone,
         time histories, realification

Features

• DPIM implementation — Direct Parametrisation of Invariant Manifolds (Cabré, Fontich & de la Llave 2003; Opreni, Vizzaccaro, Touzé & Frangi 2021)
• N-th order ODEs — native support for second-order (and higher) structural ODE systems
• External forcing — polynomial external forcing handled autonomously in the invariance equation
• Resonance handling — graph-style, complex/real normal form, and condition-number strategies
• Polynomial framework — multiindex sets, dense polynomials, and realification utilities
• FEM-agnostic — interfaces with Gridap.jl, Ferrite.jl, or any custom FEM backend
• Julia-native — multiple dispatch and static type parameters for performance

Module Overview

Status

Pre-Alpha: The API may change significantly between versions.

See Project Overview for design decisions, roadmap, and planned features.