I’ve been studying multiscale methods in the context of composite damage modeling, and I’ve come across varying classifications in the literature. In particular, the terms sequential, hierarchical, and concurrent are sometimes used distinctly, while in other cases, sequential and hierarchical appear to be treated interchangeably. My understanding is that there is a subtle difference between sequential and hierarchical methods, but I am not sure whether that is enough for a separate classification or not.
Based on my current understanding:
Sequential multiscale method: In the sequential (or one-way) multiscale method, models at different scales are solved independently, with information passed unidirectionally from the lower scale (e.g., microscale) to the higher scale (e.g., macroscale). Typically, a microscale model (e.g., a representative volume element, RVE, of fiber and matrix) is analyzed first to derive effective material properties or damage parameters. These homogenized properties are then used as inputs for a macroscale model to predict structural behavior.
Example: A microscale RVE simulation might quantify matrix cracking under tensile loading, providing a degraded stiffness tensor. This tensor is then used in a macroscale finite element model of a composite laminate to predict global failure.
Hierarchical multiscale method: In the hierarchical multiscale method, information flows both from the microscale to the macroscale (bottom-up) and from the macroscale to the microscale (top-down), but in a decoupled way. The term ‘hierarchical multiscale method’ primarily describes the nested organization of modeling scales (e.g., micro, meso, macro) inherent in composite materials, where each scale is represented by distinct models. In composite damage modeling, this means microscale damage (e.g., fiber-matrix debonding) influences macroscale properties, while macroscale stress/strain fields guide microscale simulations, often by querying pre-computed data or activating specific damage models.
Example : Consider a multi-ply composite laminate where each ply is modeled at the mesoscale, and the entire structure at the macroscale. Microscale RVE simulations are performed offline under various loading conditions and damage states to generate a comprehensive material database (e.g., stress-strain curves with stiffness degradation due to matrix cracking, fiber fracture, etc.). During a macroscale simulation of the laminate, the local stress/strain state in a ply (from the macro-model) is used to query this pre-computed database. The database then provides the appropriate degraded stiffness or strength values for that ply, effectively updating the macroscale response based on microscale damage mechanisms, without performing real-time microscale simulations for every integration point.
Concurrent multiscale method: The concurrent multiscale method solves models at multiple scales simultaneously within a single computational framework. In composite damage modeling, this involves directly embedding microscale models (e.g., RVEs or subdomains with detailed fiber-matrix geometry) into specific regions of a macroscale model where damage is critical. The microscale and macroscale domains are solved concurrently, with continuous exchange of boundary conditions and material responses.
Example : In a composite panel with a hole under tensile loading, a concurrent model might use a fine mesh around the hole to capture microscale damage (e.g., matrix cracking, fiber pullout) while using a coarse mesh elsewhere for the macroscale response. The two regions are solved together, with microscale damage directly influencing the global stress field.
I wanted to ask your opinion on whether it’s appropriate to treat sequential, hierarchical, and concurrent as three distinct categories, or whether it is more accurate to group sequential and hierarchical together as a single class of decoupled/non-concurrent methods, and then concurrent as a coupled method. An additional class can be introduced as well as a hybrid multiscale method, which I have not discussed here.
I’d greatly appreciate your perspective, especially regarding how this distinction is interpreted within the field of composite mechanics.
Thank you very much for your time.