Sample-based Design of Functionally Graded Material Structures

Spatial variation of material structures is a principal mechanism for creating and controlling spatially varying material properties in nature and engineering. While the spatially varying homogenized properties can be represented by scalar and vector fields on the macroscopic scale, explicit microscopic structures of constituent phases are required to facilitate the visualization, analysis and manufacturing of functionally graded material (FGM). The challenge of FGM structure modeling lies in the integration of these two scales. We propose to represent and control material properties of FGM at macroscale using the notion of material descriptors which include common geometric, statistical, and topological measures, such as volume fraction, correlation functions and Minkowski functionals. At microscale, the material structures are modeled as Markov random fields: we formulate the problem of design and (re)construction of FGM structure as a process of selecting neighborhoods from a reference FGM, based on target material descriptors fields. The effectiveness of the proposed method in generating a spatially varying structure of FGM with target properties is demonstrated by two examples: design of a graded bone structure and generating functionally graded lattice structures with target volume fraction fields.

Xingchen Liu, Vadim Shapiro.
2016 ASME International Design and Engineering Technical Conference, IDETC/CIE 2016, Charlotte, North Carolina
Homogenization of material properties in additively manufactured structures

Additive manufacturing transforms material into three-dimensional parts incrementally, layer by layer or path by path. Subject to the build direction and machine resolution, an additively manufactured part deviates from its design model in terms of both geometry and mechanical performance. In particular, the material inside the fabricated part often exhibits spatially varying material distribution (heterogeneity) and direction dependent behavior (anisotropy), indicating that the design model is no longer a suitable surrogate to consistently estimate the mechanical performance of the printed component.
We propose a new two-stage approach to modeling and estimating effective elastic properties of parts fabricated by fused deposition modeling (FDM) process. First, we construct an implicit representation of an effective mesoscale geometry-material model of the printed structure that captures the details of the particular process and published material information. This representation of mesoscale geometry and material of the printed structure is then homogenized at macroscale through a solution of an integral equation formulated using Green’s function. We show that the integral equation can be converted into a system of linear equations that is symmetric and positive definite and can be solved efficiently using conjugate gradient method and Fourier transform. The computed homogenized properties are validated by both finite element method and experiment results. The proposed two-stage approach can be used to estimate other effective material properties in a variety of additive manufacturing processes, whenever a similar effective mesoscale geometry-material model can be constructed.

Xingchen Liu, Vadim Shapiro.
Computer-Aided Design, Volume 78, September 2016, Pages 71-82
Efficient 3D analysis of laminate structures using ABD-equivalent
material models

Laminate composites are widely used in automotive, aerospace, and increasingly in consumer industries, due to their reduced weight and superior structural properties. However, structural analysis of complex laminate structures remains challenging. 2D finite element methods based on plate/shell theories may be accurate and efficient, but they generally do not apply to the whole structure and require identification and preprocessing of the regions where the underlying assumptions hold. Fully automated structural analysis using solid 3D elements with sufficiently high order basis functions is possible in principle, but is rarely practiced due to the significant increase in the cost of computational integration over a large number of laminate plies. We propose a procedure to replace the original laminate by much simpler new virtual material models. These virtual material models, under the usual assumptions made in lamination theory, have the same constitutive relationship as the corresponding 2D plate model of the original laminate, but require only a small fraction of computational integration costs in 3D FEA. We describe implementation of 3D FEA using these material models in a meshfree system using second order B-spline basis functions. Finally, we demonstrate their validity by showing agreement between computed and known results for standard problems.

Goldy Kumar, Vadim Shapiro.
Finite Elements in Analysis and Design, Vol. 106, 2015, pages 41-55
Random Heterogeneous Materials via Texture Synthesis

Computer models of random heterogeneous materials are becoming increasingly important in order to support the latest advances in material science, biomedical applications and manufacturing. Such models usually take the form of a microstructure whose geometry is reconstructed from a small material sample, an exemplar. A widely used traditional approach to material reconstruction relies on stochastic optimization to approximate the material descriptors of the exemplar, such as volume fraction and two-point correlation functions. This approach is computationally intensive and is limited to certain types of isotropic materials. The current invention show that formulating material reconstruction as a Markov Random Field (MRF) texture synthesis leads to a number of advantages over the traditional optimization based approaches. These include improved computational efficiency, preservation of many material descriptors, including correlation functions and Minkowski functionals, ability to reconstruct anisotropic materials, and direct use of the gray-scale material images and two dimensional cross-sections. Quantifying the quality of reconstruction in terms of correlation functions as material descriptors suggests a systematic procedure for selecting a size of neighborhood, a key parameter in the texture synthesis procedure. We support our observations by experiments using implementation with Gaussian pyramid and periodic boundary conditions.

Xingchen Liu, Vadim Shapiro.
Computational Material Science, Vol. 99, March 2015, pages 177-189
Geometric Interoperability Via Queries

The problem of geometric (model and system) interoperability is conceptualized as a non-trivial generalization of the problem of part interchangeability in mechanical assemblies. Interoperability subsumes the problems of geometric model quality, exchange, and interchangeability, as well as system integration. Until now, most of the interoperability proposals have been data-centric. Instead, we advocate a query-centric approach that can deliver interoperable solutions to many common geometric tasks in computer aided design and manufacturing, including model acquisition and exchange, metrology, and computer aided design/analysis integration.

Christoph Hoffmann, Vadim Shapiro, Vijay Srinivasan.
Computer-Aided Design, Volume 46, January 2014, Pages 148–159
Linear Algebraic Representation For Topological Structures

With increased complexity of geometric data, topological models play an increasingly important role beyond boundary representations, assemblies, finite elements, image processing, and other traditional modeling applications. While many graph- and index-based data structures have been proposed, no standard representation has emerged as of now. Furthermore, such representations typically do not deal with representations of mappings and functions and do not scale to support parallel processing, open source, and client-based architectures. We advocate that a proper mathematical model for all topological structures is a (co)chain complex: a sequence of (co)chain spaces and (co)boundary mappings. This in turn implies all topological structures may be represented by a collection of sparse matrices. We propose a Linear Algebraic Representation (LAR) scheme for mod 2 (co)chain complexes using CSR matrices and show that it supports a variety of topological computations using standard matrix algebra, without any overhead in space or running time. A full open source implementation of LAR is available and is being used for a variety of applications.

Antonio DiCarlo, Alberto Paoluzzi, Vadim Shapiro.
Computer-Aided Design, Volume 46, January 2014, Pages 269–274
Reduced Material Model of Composite Laminates For 3D Finite Element Analysis

Laminate composites are widely used in automotive, aerospace, medical, and increasingly in consumer industries, due to their reduced weight, superior structural properties and cost-effectiveness. However, structural analysis of complex laminate structures remains challenging. 2D finite element methods based on plate and shell theories may be accurate and efficient, but they generally do not apply to the whole structure, and require identification and preprocessing (dimensional reduction) of the regions where the underlying assumptions hold. Differences in and limitations of theories for thin/thick plates and shells further complicate modeling and simulation of composites. Fully automated structural analysis using 3D elements with sufficiently high order basis functions is possible in principle, but is rarely practiced due to significant increase in computational integration cost in the presence of large number of laminate plies.
We propose to replace the actual layup of the laminate structure by a simplified material model, allowing to substantially reduce the computational cost of 3D FEA. The reduced model, under the usual assumptions made in lamination theory, has the same constitutive relationship as the corresponding 2D plate model of the original laminate, but requires only a small fraction of computational integration costs in 3D FEA. We describe implementation of 3D FEA using the reduced material model in a meshfree system using second order B-spline basis functions. Finally, we demonstrate its validity by showing agreement between computed and known results for standard problems.

G. Kumar, V. Shapiro.
2012 ASME International Design and Engineering Technical Conference, IDETC/CIE 2014, Buffalo, New York
Adaptively Weighted Numerical Integration over Arbitrary Domains

In adaptively weighted numerical integration, for a given set of quadrature nodes, order and domain of integration, the quadrature weights are obtained by solving a system of suitable moment fitting equations in least square sense. The moments in the moment equations are approximated over a simplified domain that is homeomorphic to the original domain, and then are corrected for the deviation from the original domain using shape sensitivity analysis. Using divergence theorem, the moments reduce to integrals over the boundary of the simplified domain. The proposed approach supports accurate and efficient computation of quadrature weights for integration of a priori unknown functions over arbitrary 2D and 3D solid domains. Experimental results (2D) indicate that adaptively weighted integration compares favorably with more traditional approaches. Because the adaptively weighted integration avoids excessive domain subdivision, it is useful in many applications and meshfree analysis in particular.

V. Thiagarajan, V. Shapiro.
Computers & Mathematics with Applications, Volume 67, 2014.
Solving Inverse Configuration Space Problems By Adaptive Sampling

Given two shapes in relative motion, an important class of inverse configuration problems are solved by determining relative configurations that maintain set-inclusion relationships (non-interference, containment, or contact) between the shapes. This class of inverse problems includes the well-known problem of constructing a configuration space obstacle, as well as many other problems in computational design such as sweep decomposition, accessibility analysis, and dynamic packaging.We show that solutions to such problems may be efficiently approximated directly in the 6D configuration space SE(3) of relative motions by adaptive sampling. The proposed method relies on a well-known fact that the manifold of the group SE(3) is a Cartesian product of two manifold subgroups: the group of rotations SO(3) and the group of translations R3. This property allows generating desired configurations by combining samples that are generated in these subgroups independently and adaptively. We demonstrate the effectiveness of the proposed approach on several inverse problems including the problem of sweep decomposition that arises in reverse engineering applications.

S. Nelaturi, V. Shapiro
Proceedings of 2012 ACM Symposium on Solid and Physical Modeling (SPM2012), Dijon, France, October 2012.
Analysis of multi-material bonded assemblies on a non-conforming mesh

Bonded multi-material assemblies arise frequently in design, manufacturing, architecture, and materials design. It is a common wisdom that finite element analysis of such assemblies usually requires all components to be represented by compatible finite element meshes; application of meshfree methods in such situations is often considered problematic due to the need to impose additional interface conditions. Neither approach scales to deal with realistically complex models arising in many applications. We propose a simple extension of meshfree analysis on a non-conforming mesh for linear structural analysis of such multimaterial assemblies. The method is simple, can be implemented within most FEA packages and does not require either compatible meshing or complex interface boundary conditions. Our numerical experiments demonstrate that computed results are in good agreement with known analytical and computational results for well studied multi-material bonded assemblies (lap and butt joints). We also demonstrate application of the proposed method to realistically complex assembly of a mounted sculpture that cannot be easily analysed by other methods.

G. Kumar, V. Shapiro
2012 ASME International Design and Engineering Technical Conference, IDETC/CIE 2012, Chicago, IL
Configuration Workspaces of Series-Parallel Mechanisms

The workspace of a mechanism is the set of positions and orientations that is reachable by its end effector. Workspaces have numerous applications, including motion planning, mechanism design, and manufacturing process planning, but their representation and computation is challenging due to high dimensionality and geometric/topological complexity. We propose a new formulation of the workspace computation problem for a large class of mechanisms represented by series-parallel constraint graphs. A wide variety of allowable constraints include all lower pair, some higher pair, and non-collision constraints. We show that the workspace of such mechanisms may be computed by a constraint propagation algorithm. After the space of all rigid body motions is discretized, these operations can be efficiently implemented using the Fast Fourier Transform and a depth first search. In contrast to algebraic formulations, the proposed method assures that all configurations in the computed workspace not only satisfy pairwise constraints but can be reached without breaking and reassembling the mechanism.

B. McCarthy, M. Lysenko, V. Shapiro.
2012 ASME International Design and Engineering Technical Conference, IDETC/CIE 2012, Chicago, IL
Geometric Interoperability for Resilient Manufacturing

Resilient manufacturing systems require adaptable, trustable, and affordable solutions in model-based engineering (MBE) and platform-based engineering (PBE). Limited or poor geometric interoperability of the software supporting manufacturing and other engineering activities within the product life cycle is becoming a barrier not only for MBE but also curtails the potential benefits of PBE. Using a query-based approachthat is informed by the highly successful systems strategy of service-oriented architecture (SOA), these barriers can be overcome, suggesting the concepts of model interchangeability, interoperability, and integration.We illustrate the proposed methodology on several geometric interoperability tasks commonly arising in manufacturing.

C. Hoffmann, V. Shapiro, V. Srinivasan
Technical Report, Spatial Automation Laboratory, University of Wisconsin-Madison. Also available as a Purdue University technical report CSD TR #11-015.
Fourier Collision Detection

In this paper, we investigate a new approach to narrowphase collision detection for rigid objects based on the Fourier transform. This new collision test scales with respect to accuracy, and we are able to rigorously establish an upper bound on the error of our test relating the Hausdorfi metric to the number of Fourier coeffients used. Because our new form of the collision test is also a smooth inequality, it can be used as a holonomic unilateral constraint in many applications, such as path planning, rigid body dynamics, nesting or tool placement, replacing the need for more ad-hoc normal/contact based constraint solvers. Moreover, we also show how this constraint can be directly differentiated via Fourier multipliers with only a constant factor overhead, which leads to a simple method for constructing a Jacobian for both normal forces and rotational torques.

M. Lysenko
Technical Report, Spatial Automation Lab, University of Wisconsin-Madison
Geometric Issues in Computer Aided Design/Computer Aided Engineering Integration

The long-standing goal of computer aided design (CAD)/computer aided engineering (CAE) integration demands seamless interfaces between geometric design and engineering analysis/simulation tasks. The key challenge to this integration stems from the distinct and often incompatible roles geometric representations play, respectively, in design and analysis. This paper critically examines and compares known mesh-based and meshfree approaches to CAD/CAE integration, focusing on the basic tasks and components required for building fully integrated engineering applications. For each task, we identify the fundamental requirements and challenges and discuss how they may be met by known techniques and proposed solutions.

V. Shapiro, I. Tsukanov, A. Grishin
Journal of Computing and Information Science in Engineering, Volume 11, June 2011
Rapid Mapping and Exploration of Configuration Space

We describe a GPU-based computational platform for six-dimensional configuration mapping, which is the description of the configuration space of rigid motions in terms of collision and contact constraints. The platform supports a wide range of computations in design and manufacturing, including three and six dimensional configuration space obstacle computations, Minkowski sums and differences, packaging problems, and sweep computations. We demonstrate dramatic performance improvements in the special case of configuration space operations that determine interference-free or containment-preserving configurations between moving solids. Our approach treats such operations as convolutions in the six dimensional configuration space that are efficiently computed using the Fast Fourier Transform (FFT). The inherent parallelism of FFT algorithms facilitates a straightforward implementation of convolution on GPUs with existing and freely available libraries, making all such six dimensional configuration space computations practical, and often interactive.

S. Nelaturi, M. Lysenko, V. Shapiro
2011 ASME International Design and Engineering Technical Conference, IDETC/CIE 2011, Washington DC