A heterogeneous multiscale modeling framework for hierarchical systems of partial differential equations

Abstract

SUMMARYThis paper presents a heterogeneous multiscale method with efficient interscale coupling for scaledependent phenomena modeled via a hierarchy of partial differential equations. Physics at the global level is governed by one set of partial differential equations, whereas features in the solution that are beyond the resolution capability of the coarser models are accounted for by the next refined set of differential equations. The proposed method seamlessly integrates different sets of equations governing physics at various levels, and represents a consistent top-down and bottom-up approach to multi-model modeling problems. For the top-down coupling of equations, this method provides a variational residual-based embedding of the response from the coarser or global system equations, into the corresponding local or refined system equations. To account for the effects of local phenomena on the global response of the system, the method also accommodates bottom-up embedding of the response from the local or refined mathematical models into the global or coarser model equations. The resulting framework thus provides a consistent way of coupling physics between disparate partial differential equations by means of up-scaling and down-scaling of the mathematical models. An integral aspect of the proposed framework is an uncertainty quantification and error estimation module. The structure of this error estimator is investigated and its mathematical implications are delineated.

A heterogeneous modeling method for porous media flows

“…Also, a projection operator may be required in the discrete setting to ensure that the global Darcy solution is a well‐defined argument for the Darcy–Stokes operator on the right‐hand side of . A discussion of the implementation is given in Section 7.5 of the numerical results. Remark In the abstract framework for the hierarchically refined models presented in , the function space of global solutions is assumed to be a subset of the admissible space for the local solutions, that is, . While the LM is expected to capture the more refined physics of the system, the mathematical function spaces must satisfy the regularity requirements of the particular governing PDEs.…”

“…Remark 8: In the abstract framework for the hierarchically refined models presented in [12], the function space of global solutions is assumed to be a subset of the admissible space for the local solutions, that is, S G ⊆S L . While the LM is expected to capture the more refined physics of the system, the mathematical function spaces must satisfy the regularity requirements of the particular governing PDEs.…”

Residual-based turbulence models and arbitrary Lagrangian–Eulerian framework for free surface flows

“…The model for the fine-scale fields (31) is substituted in the coarse-scale problem (16)- (17). The resulting formulation can be written in an abstract form as:…”

Edge stabilization and consistent tying of constituents at Neumann boundaries in multi‐constituent mixture models

“…zero applied tractions. Following the line of thought in [23][24][25][26][27][28], a numerical method is developed that draws from the stabilized discontinuous Galerkin method for finite strain kinematics with an underlying Lagrange multiplier interface formulation. The derivation of the new method hinges upon a multiscale decomposition of the deformation map locally at the Neumann boundary and subsequent modeling of the fine scales via edge bubble functions.…”

“…The penalty parameter in the Nitsche method needs to be defined to ensure the coercivity of the method, and there have been many works to define this parameter through the following: (i) an a priori analysis, (ii) solving of global or local eigenvalue problems, and (iii) bubble functions approach [18][19][20][21][22]. Masud and coworkers [23][24][25][26][27][28] have developed a unified formulation for interface coupling and frictional contact modeling where the penalty parameter is derived through variational multiscale (VMS) framework, and a Lagrange multiplier field is approximated as a simple average of fluxes. Truster and Masud [29] have extended this framework to finite deformations where the stabilization tensor is consistently derived and is a function of both material and geometric nonlinearity.The deformation of multi-constituent mixtures at the Neumann boundaries requires imposing constraint conditions such that the constituents deform in a self-consistent fashion.…”