Computational Physics Group

Karel Matous










Wavelet based reduced order models for microstructural analyses

R.A. van Tuijl1, C. Harnish2, K. Matous2, J.J.C. Remmers1 and M.G.D. Geers1

1Eindhoven University of Technology, P.O. Box 513,
Eindhoven, The Netherlands

2Center for Shock Wave-processing of Advanced Reactive Materials

Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame,
Indiana 46556, USA.


    This paper proposes a novel method to accurately and efficiently reduce a microstructural mechanical model using a wavelet based discretisation. The model enriches a standard reduced order modelling (ROM) approach with a wavelet representation. Although the ROM approach reduces the dimensionality of the system of equations, the computational complexity of the integration of the weak form remains problematic. Using a sparse wavelet representation of the required integrands, the computational cost of the assembly of the system of equations is reduced significantly. This wavelet-reduced order model (W-ROM) is applied to the mechanical equilibrium of a microstructural volume as used in a computational homogenisation framework. The reduction technique however is not limited to micro-scale models and can also be applied to macroscopic problems to reduce the computational costs of the integration. For the sake of clarity, the W-ROM will be demonstrated using a one-dimensional example, providing full insight in the underlying steps taken.


   This paper presents a novel hyper-reduced method by introducing a wavelet basis and a MRA to integrate the weak form up to a pre-set tolerance. The innovative aspects of the
proposed method are:
        - The wavelet reduced integration does not require an offline calibration step using a second set of (stress-based) snapshots to construct the reduced integration scheme. This reduces the input parameters required to construct the integration scheme to the strain modes only (projected on the dyadic wavelet grid) and a tolerance to control the level of approximation.
       - The reduced integration uses the wavelet-based MRA to enable an adaptive and local refinement of the dyadic grid used to integrate the stress, internal forces and stiffness. Conversely, the adaptive dyadic grid will also coarsen when smooth functions are to be approximated.
       - The error in the integration is controlled using a pre-set tolerance. This is verified by quantifying the microfluctuation error of the W-ROM relative to the FOM. The resulting error in micro-fluctuation coefficients is typically bounded by the imposed tolerance.

    The reduction of the number of stress evaluations to perform the integration with respect to the original Gauss integration scheme used in the ROM is demonstrated and the compression is shown to be inversely proportional to the tolerance.
    Hyper-reduced models such as Empirical Interpolation Methods and Empirical Cubature Methods rely on offline determined integration points, and sometimes a basis for the integrand. This allows for a higher compression ratio, but requires a detailed sampling of the high-dimensional snapshot space including the history parameters for path dependent materials, to ensure that no physical modes are missing since the integration scheme is constructed a priori. Many hyper-reduction techniques combine the sampling of the non-linear terms, such as the stress field σ or the internal force integrand ϕ, with the sampling of the kinematics. Therefore no additional computations are required for the construction of the snapshots of the nonlinear term. The accuracy of the reduced integration of history dependent material models however, requires extensive sampling of both the kinetics and the kinematics present in the parameter space.
    The Wavelet-Reduced Order Model, on the other hand, adaptively determines the points required for accurate integration in-situ and only requires prior accurate sampling of the kinematics. The sampled kinematics are required to construct the symmetric gradient of the reduced basis ∇2R(x) and is therefore less sensitive to the sampled snapshots, and algorithmic choices to select integration points.
    Note that there is no theoretical limitation to expand W-ROM to two or three dimensional microstructural models. This 1D model provides a transparent view of all underlying principles and implementation aspects of the method proposed. Expanding the method to multiple dimensions will however require multi-dimensional adaptive wavelet transforms as presented by Paolucci et al. in [32] and more elaborate storage solutions for the internal variables and modes by approximating each variable independently on a separate wavelet grid to allow fast wavelet transforms while still limiting the memory usage.

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