Dimensional hyper-reduction of nonlinear finite element models via empirical cubature

We present a general framework for the dimensional reduction, in terms of number of degrees of freedom as well as number of integration points (“hyper-reduction”), of nonlinear parameterized finite element (FE) models. The reduction process is divided into two sequential stages. The first stage consists in a common Galerkin projection onto a reduced-order space, as well as in the condensation of boundary conditions and external forces. For the second stage (reduction in number of integration points), we present a novel cubature scheme that efficiently determines optimal points and associated positive weights so that the error in integrating reduced internal forces is minimized. The distinguishing features of the proposed method are: (1) The minimization problem is posed in terms of orthogonal basis vector (obtained via a partitioned Singular Value Decomposition) rather that in terms of snapshots of the integrand. (2) The volume of the domain is exactly integrated. (3) The selection algorithm need not solve in all iterations a nonnegative least-squares problem to force the positiveness of the weights. Furthermore, we show that the proposed method converges to the absolute minimum (zero integration error) when the number of selected points is equal to the number of internal force modes included in the objective function. We illustrate this model reduction methodology by two nonlinear, structural examples (quasi-static bending and resonant vibration of elastoplastic composite plates). In both examples, the number of integration points is reduced three order of magnitudes (with respect to FE analyses) without significantly sacrificing accuracy.Peer ReviewedPostprint (published version

Enabling parametric design space exploration by non-designers

In mass customization, software configurators enable novice end-users to design customized products and services according to their needs and preferences. However, traditional configurators hardly provide an engaging experience while avoiding the burden of choice. We propose a Design Participation Model to facilitate navigating the design space, based on two modules. Modeler enables designers to create customizable designs as parametric models, and Navigator subsequently permits novice end-users to explore these designs. While most parametric designs support direct manipulation of low-level features, we propose interpolation features to give customers more flexibility. In this paper, we focus on the implementation of such interpolation features into Navigator and its user interface. To assess our approach, we designed and performed user experiments to test and compare Modeler and Navigator, thus providing insights for further developments of our approach. Our results suggest that barycentric interpolation between qualitative parameters provides a more easily understandable interface that empowers novice customers to explore the design space expeditiously

A level-set shape metamorphosis with mechanical constraints for geometrically graded microstructures

Functional performance benefits brought by graded materials or a variation of microstructures have been established. However, the challenge of connecting the adjacent microstructures remains open. Without connected microstructures, the design is only theoretical and cannot be realized in practice. This paper presents a shape metamorphosis approach to address this challenge. The approach formulates an optimization problem to minimize the difference between two shapes, where the shapes generated during the optimization iterations form the graded microstructures. A stiffness-based constraint is proposed to eliminate potential breakage or shrinkages in generating the intermediate shapes, which are used to build a transition zone for the two shapes to be connected. This also ensures that the optimal functionality is not compromised significantly. A series of numerical studies demonstrate that the proposed approach is capable of producing smoothly connected microstructures for multiscale optimization

Dimensional hyper-reduction of nonlinear finite element models via empirical cubature

We present a general framework for the dimensional reduction, in terms of number of degrees of freedom as well as number of integration points (“hyper-reduction”), of nonlinear parameterized finite element (FE) models. The reduction process is divided into two sequential stages. The first stage consists in a common Galerkin projection onto a reduced-order space, as well as in the condensation of boundary conditions and external forces. For the second stage (reduction in number of integration points), we present a novel cubature scheme that efficiently determines optimal points and associated positive weights so that the error in integrating reduced internal forces is minimized. The distinguishing features of the proposed method are: (1) The minimization problem is posed in terms of orthogonal basis vector (obtained via a partitioned Singular Value Decomposition) rather that in terms of snapshots of the integrand. (2) The volume of the domain is exactly integrated. (3) The selection algorithm need not solve in all iterations a nonnegative least-squares problem to force the positiveness of the weights. Furthermore, we show that the proposed method converges to the absolute minimum (zero integration error) when the number of selected points is equal to the number of internal force modes included in the objective function. We illustrate this model reduction methodology by two nonlinear, structural examples (quasi-static bending and resonant vibration of elastoplastic composite plates). In both examples, the number of integration points is reduced three order of magnitudes (with respect to FE analyses) without significantly sacrificing accurac

Sparse data driven mesh deformation

Example-based mesh deformation methods are powerful tools for realistic shape editing. However, existing techniques typically combine all the example deformation modes, which can lead to overfitting, i.e. using an overly complicated model to explain the user-specified deformation. This leads to implausible or unstable deformation results, including unexpected global changes outside the region of interest. To address this fundamental limitation, we propose a sparse blending method that automatically selects a smaller number of deformation modes to compactly describe the desired deformation. This along with a suitably chosen deformation basis including spatially localized deformation modes leads to significant advantages, including more meaningful, reliable, and efficient deformations because fewer and localized deformation modes are applied. To cope with large rotations, we develop a simple but effective representation based on polar decomposition of deformation gradients, which resolves the ambiguity of large global rotations using an as-consistent-as-possible global optimization. This simple representation has a closed form solution for derivatives, making it efficient for our sparse localized representation and thus ensuring interactive performance. Experimental results show that our method outperforms state-of-the-art data-driven mesh deformation methods, for both quality of results and efficiency