Multiscale finite element musculoskeletal model for intact knee dynamics

In this paper, we present a new approach to model the steady-state heat transfer in heterogeneous materials. The multiscale finite element method (MsFEM) is improved and used to solve this problem. MsFEM is a fast and flexible method for upscaling. Its numerical efficiency is based on the natural parallelization of the main computations and their further simplifications due to the numerical nature of the problem. The approach does not require the distinct separation of scales, which makes its applicability to the numerical modeling of the composites very broad. Our novelty relies on modifications to the standard higher-order shape functions, which are then applied to the steady-state heat transfer problem. To the best of our knowledge, MsFEM (based on the special shape function assessment) has not been previously used for an approximation order higher than p = 2, with the hierarchical shape functions applied and non-periodic domains, in this problem. Some numerical results are presented and compared with the standard direct finite-element solutions. The first test shows the performance of higher-order MsFEM for the asphalt concrete sample which is subject to heating. The second test is the challenging problem of metal foam analysis. The thermal conductivity of air and aluminum differ by several orders of magnitude, which is typically very difficult for the upscaling methods. A very good agreement between our upscaled and reference results was observed, together with a significant reduction in the number of degrees of freedom. The error analysis and the p-convergence of the method are also presented. The latter is studied in terms of both the number of degrees of freedom and the computational time.

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An Online Generalized Multiscale Finite Element Method for Unsaturated Filtration Problem in Fractured Media

Mathematics ◽

10.3390/math9121382 ◽

2021 ◽

Vol 9 (12) ◽

pp. 1382

Author(s):

Denis Spiridonov ◽

Maria Vasilyeva ◽

Aleksei Tyrylgin ◽

Eric T. Chung

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Model Reduction ◽

Basis Functions ◽

Computational Domain ◽

Filtration Problem ◽

Spectral Problems ◽

Multiscale Finite Element Method ◽

Multiscale Finite Element ◽

Element Method

In this paper, we present a multiscale model reduction technique for unsaturated filtration problem in fractured porous media using an Online Generalized Multiscale finite element method. The flow problem in unsaturated soils is described by the Richards equation. To approximate fractures we use the Discrete Fracture Model (DFM). Complex geometric features of the computational domain requires the construction of a fine grid that explicitly resolves the heterogeneities such as fractures. This approach leads to systems with a large number of unknowns, which require large computational costs. In order to develop a more efficient numerical scheme, we propose a model reduction procedure based on the Generalized Multiscale Finite element method (GMsFEM). The GMsFEM allows solving such problems on a very coarse grid using basis functions that can capture heterogeneities. In the GMsFEM, there are offline and online stages. In the offline stage, we construct snapshot spaces and solve local spectral problems to obtain multiscale basis functions. These spectral problems are defined in the snapshot space in each local domain. To improve the accuracy of the method, we add online basis functions in the online stage. The construction of the online basis functions is based on the local residuals. The use of online bases will allow us to get a significant improvement in the accuracy of the method. We present results with different number of offline and online multisacle basis functions. We compare all results with reference solution. Our results show that the proposed method is able to achieve high accuracy with a small computational cost.

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Image-based semi-multiscale finite element analysis using elastic subdomain homogenization

Meccanica ◽

10.1007/s11012-021-01378-4 ◽

2021 ◽

Author(s):

J. Jansson ◽

K. Salomonsson ◽

J. Olofsson

Keyword(s):

Finite Element ◽

Analytical Method ◽

Effective Properties ◽

Reference Model ◽

Computational Time ◽

Component Level ◽

Element Analysis ◽

Model Fidelity ◽

Multiscale Finite Element ◽

Anisotropic Elastic

AbstractIn this paper we present a semi-multiscale methodology, where a micrograph is split into multiple independent numerical model subdomains. The purpose of this approach is to enable a controlled reduction in model fidelity at the microscale, while providing more detailed material data for component level- or more advanced finite element models. The effective anisotropic elastic properties of each subdomain are computed using periodic boundary conditions, and are subsequently mapped back to a reduced mesh of the original micrograph. Alternatively, effective isotropic properties are generated using a semi-analytical method, based on averaged Hashin–Shtrikman bounds with fractions determined via pixel summation. The chosen discretization strategy (pixelwise or partially smoothed) is shown to introduce an uncertainty in effective properties lower than 2% for the edge-case of a finite plate containing a circular hole. The methodology is applied to a aluminium alloy micrograph. It is shown that the number of elements in the aluminium model can be reduced by $$99.89\%$$ 99.89 % while not deviating from the reference model effective material properties by more than $$0.65\%$$ 0.65 % , while also retaining some of the characteristics of the stress-field. The computational time of the semi-analytical method is shown to be several orders of magnitude lower than the numerical one.

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