The geometrical parameter bounds are selected to facilitate access during the optimization procedure to significant variations in both Poisson’s ratio and axial stiffness. Future studies should seek to experimentally validate the multiscale geometries presented within this paper. In addition, it would be interesting to explore higher-order homogenization schemes for the multiscale structural analysis, with the aim to exploit instabilities such as buckling at both scales to realize larger deformations. Multiscale topology optimization frameworks permit the microscale description of a structure to be spatially optimized to fulfill functional objectives at the macroscale. This permits hierarchical structures with superior mechanical (Imediegwu et al. 2019; Murphy et al. 2021b), thermal (Imediegwu et al. 2021), and dynamic (Nightingale et al. 2021) performance to be generated relative to conventional topology optimization-based approaches.
1 Microscale model
In a two-level setup, at any macro time step ormacro iteration step, the procedure is as follows. The idea is to decompose the wholecomputational domain into several overlapping or non-overlappingsubdomains and to obtain the numerical solution over the whole domainby iterating over the solutions on these subdomains. The domaindecomposition method is not limited to multiscale problems, but it canbe used for multiscale problems. This is a way of summing up longrange interaction potentials for a large set of particles. Thecontribution to the interaction potential is decomposed intocomponents with different scales and these different contributions areevaluated at different levels in a hierarchy of grids. These methods are certainly more accurate than their single-scale, isotropic predecessors, but fall short when trying to analyze novel parts/materials for which there is no historical correlations or empirical guide-posts.
1 Strain space exploration
- Each had different programs that tried to unify computational efforts, materials science information, and applied mechanics algorithms with different levels of success.
- Multiscale topology optimization frameworks permit the microscale description of a structure to be spatially optimized to fulfill functional objectives at the macroscale.
- By combining both viewpoints, one hopes to arrive at areasonable compromise between accuracy and efficiency.
- The primary benefit of this uniformity is the existence of rotational and reflectional symmetries, which enables up to eight unique-second Piola–Kirchhoff stress tensors to be derived from a single parent simulation.
- Various tricks are then used to entice the microscalesimulations on small domains to behave like a full simulation on thewhole domain.
Importantly, a full-factorial DOE is selected for this application, as the uniform distribution of simulation nodes affords a number of advantages. The primary benefit of this uniformity is the existence of rotational and reflectional symmetries, which enables up to eight unique-second Piola–Kirchhoff stress tensors to be derived from multi-scale analysis a single parent simulation. Symmetries are identified within the DOE by applying combinations of two-dimensional transformation matrices to a series of points representing all four structural members and all four strain direction vectors. The transformed set of points is then inspected to determine how each parameter is permuted during the transformation procedure.
Macro-micro formulations for polymer fluids
However, in the general case, the generalized Langevinequation can be quite complicated and one needs to resort toadditional approximations in order to make it tractable. While heterogeneity offers huge advantages in performance (making airplanes, space shuttles and lightweight cars possible), it also introduces difficulties in the engineering design. Presently, there is not enough computational power to include all the important details within a single Finite Element (FE) model, as is customary in industry. This is because that would require a high-resolution model too complex to be feasibly solved.
An ANN-assisted efficient enriched finite element method via the selective enrichment of moment fitting
- Multiple scientific articles were written, and the multiscale activities took different lives of their own.
- In addition, it would be interesting to explore higher-order homogenization schemes for the multiscale structural analysis, with the aim to exploit instabilities such as buckling at both scales to realize larger deformations.
- Brandt also noted thatone might be able to exploit scale separation to improve theefficiency of the algorithm, by restricting the smoothing operationsat fine grid levels to small windows and for few sweeps.
- In this instance, only 1.45% of the total strain space is resolved to compute the macroscale displacement field, even though the direct strains exceed 24%.
- This enables functional objectives cast exclusively as a function of infinitesimal displacements to be efficiently targeted without loss of accuracy (Christensen et al. 2023).
- This is a strategy for choosing thenumerical grid or mesh adaptively based on what is known about thecurrent approximation to the numerical solution.
Multiscale ideas have also been used extensively in contexts where nomulti-physics models are involved. An example of such problems involve the Navier–Stokes equations for incompressible fluid flow. E, “Stochastic models of polymeric fluids at small Deborah number,” submitted to J. This is further evidenced by the straightforward minimization of the error functional depicted in Fig. Check if you have access through your login credentials or your institution to get full access on this article. A classical example in which matched asymptotics has been used isPrandtl’s boundary layer theory in fluid mechanics.
- Precomputing the inter-atomic forces asfunctions of the positions of all the atoms in the system is notpractical since there are too many independent variables.
- In the equation-free approach, particularly patchdynamics or the gap-tooth scheme, the starting point is the microscalemodel.
- The accuracy of these results are verified using high-fidelity single-scale finite-element analysis.
- In response to these challenges, we present MuSpAn, a Multiscale Spatial Analysis package designed to provide straightforward access to both well-established and cutting-edge mathematical analysis tools.
- To verify the accuracy of the optimized macroscale displacement fields predicted by the multiscale structural analysis, high-fidelity simulations of the reconstructed geometries are performed.
- To reduce the computational cost of expensive nonlinear structural analysis, the present framework is restricted to two-dimensional space; however, only trivial modifications would be required to extend the present framework to the three-dimensional regime.
These slowly varying quantities aretypically the Goldstone modes of the system. For example, the densities ofconserved quantities such as mass, momentum and energy densities areGoldstone modes. The equilibrium states of macroscopicallyhomogeneous systems are parametrized by the values of thesequantities.
