What are the best practices for coupling different numerical methods across scales?

Multiscale problems are complex issues that involve interactions and processes at multiple levels of organization or observation. These problems often require the consideration of different spatial and temporal scales and the integration of information and data from various disciplines. Multiscale problems can arise in fields such as biology, physics, engineering, and environmental science, and they often require innovative approaches and techniques to be effectively addressed.

Multiscale problems are challenging because they involve interactions and processes at multiple levels of organization, making them inherently complex. These problems often require the integration of information and data from different disciplines, and the consideration of various spatial and temporal scales. Additionally, addressing multiscale problems may require innovative approaches and techniques to effectively capture the complexity and

Es gibt auch Ans?tze, bei denen die Finite-Elemente-Methode als quadratische Variante verwendet wird. Zuerst wird das makroskopische Problem mit Finiten Elementen diskretisiert. Danach gibt es in jedem Quadratur-Punkt ein lokales mikroskopisches Finite-Elemente-Verfahren. Allerdings konvergieren die Diskretisierungsparameter nicht in jedem Fall.

in my mind ,tackling multiscale problems with computational methods involves developing models that can capture interactions and processes across different scales. This may involve using techniques such as hierarchical modeling, agent-based modeling, or hybrid models that integrate different computational approaches. Additionally, utilizing high-performance computing resources can help in simulating complex multiscale systems and analyzing large datasets. Furthermore, the development of algorithms that can efficiently handle multiscale data and interactions is crucial in addressing these challenging problems.

I strongly belive that the best practices for coupling different numerical methods across scales involve ensuring consistency and accuracy in the transfer of information between the different scales. This may include using interpolation and averaging techniques to bridge the gap between scales, as well as verifying the compatibility of the methods at the interfaces. Additionally, maintaining stability and convergence across the different scales is crucial, and employing adaptive mesh refinement or time-stepping strategies can help in efficiently capturing multiscale phenomena. Regular validation and verification of the coupled methods are also essential to ensure the reliability of the results.

Coupling different numerical methods across scales is a cost-effect approach to develop and optimise wood-based products. It was described with specific applications in Chapter 4.3 of the latest published fisrt-of-its-kind Modelling Guide for Timber Structures (web.fpinnovations.ca/modelling).

One example of coupling different numerical methods across scales is the use of multiscale finite element methods, which combine macroscopic and microscopic models to capture the behavior of materials at different scales. Another example is the use of hybrid models in computational fluid dynamics, where continuum models are coupled with molecular dynamics simulations to capture fluid behavior at both macroscopic and molecular scales. Additionally, in climate modeling, different numerical methods are often coupled to simulate interactions between atmospheric, oceanic, and land surface processes at various spatial and temporal scales.

In my experience, strongly coupled FVMFEM is the best choice for fluid-solid interactions. In which flow is discretized using FVM and structure by FEM. They solved the governing equations and exchange the data necessary to each other by strongly coupled mechanisms.