Zébulon

Nonlinear Finite Element Solver

Zébulon is the flagship finite element solver within the Z-set Suite, designed to tackle the full spectrum of problems in structural mechanics. From the outset, its focus has been on highly nonlinear material behaviors, and over time its scope has expanded to include thermal and diffusion processes, as well as their couplings with mechanical problems.

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Material Nonlinearity

Material models are seamlessly integrated through Z-mat, ranging from simple isotropic elasticity to complex anisotropic elasto-viscoplasticity, polymers, damageable composites, and multiscale formulations. Nonlinearities related to materials, large deformations, or contact/friction are treated with a comprehensive set of solvers, offering users detailed control over both physical and numerical convergence.

Thermal, Diffusion, and Coupled Analyses

Zébulon includes a thermal analysis module capable of solving both steady-state and transient thermal problems. It also handles multiphase diffusion (Fick’s law) and can address other elliptic or parabolic equations in a similar manner.
A weak coupling framework allows users to combine multiple fields (e.g., thermo-mechanical-diffusion simulations). For more advanced multiphysics needs, Zébulon also supports external coupling with specialized solvers, such as fluid dynamics or radiation software.

Crack Simulation

The Z-cracks module enables accurate modeling of crack initiation and propagation. By combining remeshing techniques with robust SIF calculation approaches, Z-cracks offers a robust and efficient solution for fatigue-related or brittle fracture problems.

Static, Dynamic, and Modal Analysis

With its implicit solver, Zébulon can address both static and transient dynamic simulations.
It also provides modal analysis through several eigenvalue solvers, making it possible to investigate vibration modes and structural dynamics.

High-Performance Computing

Zébulon is optimized for modern HPC environments using a two-level parallelization strategy:

• On shared-memory systems, multithreading speeds up both material integration and the resolution of linear systems.
• On distributed-memory clusters, domain decomposition methods are implemented for large-scale simulations.

User Extensibility

The solver can be customized and extended in two main ways:

• Z-program, an interpreted scripting language inspired by C++, giving access to internal Z-set objects.
• A plug-in mechanism, based on object-oriented inheritance, allowing new boundary conditions, element types, post-processing features, or numerical procedures to be directly integrated into the solver.

Element Library

General-purpose elements:

• 2D: linear and quadratic triangles/quadrilaterals
• 3D: tetrahedra, hexahedra, prisms, pyramids (linear & quadratic)
• 1D and 1.5D elements
• Structural elements: beams, trusses, shells
• Layered elements for laminates
• Interface elements with cohesive zone models

Specialized elements:

• Periodic elements
• Bubble elements for incompressible materials
• Non-local and micromorphic elements

Formulations:

• Plane stress, plane strain, axisymmetric
• Small or finite strains
• Generalized plane strain (2.5D)
• Cosserat and micromorphic continua

Resolution Algorithms

• Nonlinear solvers: Quasi-Newton, BFGS option, arc-length methods for buckling and softening.
• Time integration: Adaptive time-stepping, implicit θ-method, explicit Runge-Kutta with local step control.
• Linear solvers: Direct and iterative, optimized for large-scale parallel systems.

Model Reduction

Zébulon integrates innovative strategies to accelerate nonlinear problem solving, such as:

• Hyper-reduction techniques
• Skip-cycle acceleration methods
• Explicit schemes for stabilized cycle convergence

Adaptive Remeshing & Crack Growth

• Automatic 2D/3D adaptive mesh refinement
• Field transfer methods for remeshed domains
• Stress intensity factors via the G-θ method
• Automatic crack insertion and propagation (planar & non-planar)

Multiscale and Multiphysics Capabilities

• FE² framework: micro-scale FE models define behavior at macro-scale integration points.
• Submodeling and Arlequin methods for localized refinement.
• Internal and external code coupling for fluid–structure and other multiphysics applications.
• Homogenization approaches with specialized boundary conditions.

External Parameters

Zébulon supports an unlimited number of external parameters (uniform or spatially varying), with import/export options from preliminary analyses for seamless multiphysics workflows.