Geometric modeling: Challenges for Additive Manufacturing, Design and Analysis

Locally Refined B-splines (LRB) represents a multivariate generalization of knot insertion for univariate spline spaces. For LRB refinement is through sequential insertion of axes parallel mesh line segments. For each refinement step the mesh-line segment must divide into two disjoint regions the support of at least one tensor product (TP) B-spline from the collection of TP B-splines spanning the spline space. In this presentation we propose a new subclass of LRB. It performs better than Truncated Hierarchical B-splines when both methods are defined over the same mesh of knotlines: 1. The spline space defined by the knotlines are always filled which is not always the case for THB; 2. The LRB basis functions have better scaling factors than for THB, resulting in better condition numbers of both the mass and stiffness matrices. The minimal refinements regions of this approach will be slightly longer or wider than the minimal refinement for THB to ensure that the support of at least one TP B-splines is split. The resulting TP B-splines for this subclass are guaranteed to be linearly independent.

The needs of modern (additive) manufacturing (AM) technologies can be satisfied no longer by boundary representations (B-reps), as AM enables the manipulation and fabrication of interior (graded) materials as well as porosity. Further, while the need for a tight coupling between design and analysis has been recognized as crucial almost since geometric modeling (GM) was conceived, contemporary GM systems only offer a loose link between the two, if at all. For about half a century, since the 70’s, (trimmed) Non Uniform Rational B-spline (NURBs) surfaces have been the B-rep of choice for virtually all of the GM industry. Fundamentally, B-rep GM has evolved little during this period and is no longer able to fulfill the needs of modern (additive) manufacturing, namely heterogeneity and lattice/porosity support. In this talk, we seek to examine an extended (trimmed) NURBs volumetric representation (V-rep) that successfully confronts the existing and anticipated design, analysis, and manufacturing foreseen challenges. We extend all fundamental B-rep GM operations, such as primitive and surface constructors and Boolean operations, to trimmed trivariate V-reps. This enables the much-needed tight link to (Isogeometric) analysis (IGA) on one hand and the full support of (porous, heterogeneous, and anisotropic) modern/additive manufacturing needs on the other.

Surface stackability is the measure of how well a freeform surface can be decomposed into stackable components. Under this view, a freeform surface is treated as one modality of a single bimodal geometric object which adheres to two configurations: the deployed surface state and the stacked volume state. This dual configuration introduces a geometric link between freeform surfaces and volume foliations.

Stackability has applications in freeform architecture and digital fabrication methods, such as hot blade cutting and conformal 3D printing, and allows for efficiency in material use, packing, storage and transportation.

I presented work-in-progress results and discussed possible future research questions.

In this talk, we combine computer aided geometric design methods with deep learning technologies. The final objective is to develop and apply adaptive data fitting schemes for the design of data-driven free-form spline geometries. Depending on the acquisition process, the nature of the data can strongly vary, from uniformly distributed to scattered and affected by noise; yet the reconstructed geometric models are required to be compact, highly accurate, and smooth, while simultaneously capturing key geometric features. We propose data-driven parameterization methods based on (geometric) deep learning, considering either structured or unstructured point cloud configurations. On the basis of this learning model, we introduce THB-spline fitting schemes with moving parameterization and present a selection of numerical examples.

An algebra is a vector space A together with a rule for multiplication A×A→A. The real numbers, complex numbers, quaternions, dual quaternions, and conformal quaternions are examples of algebras with potential applications in geometric modeling. The goal of this talk is to show how to generate new algebras from known algebras doubling the size of the algebra by adjoining one new element along with new multiplication rules characterizing how this new element interacts with preexisting elements of the algebra. Examples are provided to illustrate the method. The connection to even dimensional subalgebras of Clifford algebras is also explained. We will close with some open questions for future research.

The functionality and the quality of surfaces are important topics in engineering.How can we visualize these aspects in a preprocessing step during simulations? A surface is under pressure and deforming. Is it bending without or with deforming the surface-metric? Mathematical concepts to deal with these kind of problems are differential geometry and infinitesimal bendings.Shadow-curves are an intuitive visualization tool. We show in this talk that as long as the shadow-lines stay stationary during the deformation the surface is infinitesimal rigid.

Subdivision schemes are used to generate smooth curves by iteratively refining an initial control polygon. The simplest such schemes are corner-cutting schemes, which specify two distinct points on each edge of the current polygon and connect them to get the refined polygon, thus cutting off the corners of the current polygon. While de Boor [1] showed that this process always converges to a Lipschitz continuous limit curve, no matter how the points on each edge are chosen, Gregory and Qu [2] discovered that the limit curve is differentiable under certain constraints. We extend this result and show that the limit curve can even be curvature continuous for a specific sequence of cut ratios.

Moving curves that follow a rational curve and moving surfaces that follow a rational surface serve as a bridge between the parametric forms and implicit forms. Their algebraic counterparts are special syzygies of the parametric equations of rational curves or surfaces. Over the past thirty years, the technique of moving curves and moving surfaces have been proven to be significant in solving many important problems in geometric modeling, such as fast implicitization, intersection computation, singularity computation, reparametrization as well as providing easy inversion formulas for points. We review the state-of-the-art results in µ-bases theory and applications for rational curves and surfaces, and raise unsolved problems for future research.

Finding parameterizations of spatial point data is a fundamental step for surface reconstruction in Computer Aided Geometric Design. Especially the case of unstructured point clouds is challenging and not widely studied. In this work, we show how to parameterize a point cloud by using barycentric coordinates in the parameter domain, with the aim of reproducing the parameterizations provided by quadratic triangular Bézier surfaces. To this end, we train an artificial neural network that predicts suitable barycentric parameters for a fixed number of data points. In a subsequent step we improve the parameterization using non-linear optimization methods. We then use a number of local parameterizations to obtain a global parameterization using a new overdetermined barycentric parameterization approach. We study the behavior of our method numerically in the zero-residual case (i.e., data sampled from quadratic polynomial surfaces) and in the non-zero residual case and observe an improvement of the accuracy in comparison to standard methods.

In engineering, the applications of machine learning and deep learning models have predominantly focused on reducing the computational costs of simulations by creating low-fidelity surrogate models that predict performance nearly instantly. Until recently, the capability of these models to generate innovative solutions was limited. The advent of modern generative models (GMs) and their application in engineering have significantly altered the landscape by enabling the creation of innovative shapes in addition to performance prediction.

However, as with these models, both the input and output streams consist of low-level 3D shape representations, they fail to capture structural and shape characteristics that are essential for performance analysis. Consequently, common issues in the generated designs include lack of surface smoothness and a large number of invalid designs, such as non-watertight or self-intersecting designs. This is particularly critical in engineering analysis where maintaining surface smoothness and validity is vital, as even minor local variations in surface quality can significantly impact performance. Additionally, due to their unsupervised nature, these models fail to incorporate any notion of physics.

In this work, we propose the use of physics-informed geometric operators (GOs) to enrich the geometric data provided to the employed GMs. We claim that this addition enables the extraction of useful high-level shape characteristics, even when using simple model architectures or low-level data representations like design parameters. GOs leverage the shape’s differential and/or integral properties – retrieved via Fourier analysis, curvature, geometric moments, and their invariants – thereby introducing high-level intrinsic geometric information and physics into the resulting shape descriptors. These operators capture both global and local shape features by explicitly encoding the relevant shape information. This not only augments the training dataset with a compact geometric representation of free-form shapes but also embeds physical information. The latter is achieved through the selection of shape characteristics that correlate with the design’s performance metrics (such as wave-making resistance, lift, and drag), thereby making the model physics-informed. Our experiments further demonstrate that GO-augmented shape descriptors result in measurable improvements in modelling accuracy and enhancements in the model’s generalisation capability. Various finite combinations of GO-induced values can capture different sets of underlying geometric information, which are studied here for their efficacy in acting as sufficient and compact signatures of the corresponding shape surfaces.

With the increasing need of autonomous robots for complicated environment situation such as underwater application, more robust algorithm is needed. In simultaneous localization and mapping algorithm, one of the core parts is the back end. The noisy measurement and robot trajectory are process to correct the drifting error using loop closure measurement (recognition of previously visited place). The process of optimizing the robot poses with respect to the sensor measurements is called pose graph optimization (PGO). Solving a PGO problem is equivalent to solve a maximum likelihood estimation problem where the objective function is the error between the measurement and the poses. The classical framework is to use a least-square formulation. However, this formulation has several drawbacks: The sensitivity to poses initialization first can lead to a local minima solution as it is a nonconvex problem. Then the presence of wrong measurement with large error, also called outliers, can lead to arbitrary wrong solution. In this research, we aim at studying a method for PGO which leverages the problem of initialization and is robust to outliers’ presence. The initialization sensitivity problem comes from the nonconvexity of the minimization problem as it introduces multiple local minima. The proposed solution is to relax the problem into a convex one with a single global minimum. The solution of the relaxed problem can be reprojected on the initial nonconvex problem feasible set. Additionally, using this method we have a contract on the certifiability of our solution, i.e we can ensure that the solution is the global minima, or we detect the failure. For the sensitivity to outliers, it mainly comes from the fact that the ii formulation is quadratic in the error terms so if one measurement contains a wrong large error it will dominate the objective function. The proposed approach here is the use of M-estimator. A M-estimator is adding a loss function around the error term to mitigate its impact if it is too large. This thesis aims at comparing different loss function that can be used on the chosen convex relaxation approach. Additionally, we suppose that only edge which are loop closure can be outliers. After deriving the formulation corresponding to our choice, we test on 3 synthetic datasets the different loss function and compare them. Our results show that the convex loss function, i.e $L^1,L^2$, identity tested here do well for highly connected pose graph but failed to stay robust for low connectivity pose graph.

Sensor Data is diverse. Examples are pictures recorded by CMOS sensors common in any cell-phone, point clouds recorded by laser scanners, computed tomography scans as in medical imaging, recordings of vibrations by accelerometers, or pressure waves recorded by piezoelectric elements.

The presentation will give an insight on how to use such data for the mechanical analysis of structures [1, 2]. It finishes by presenting a particularly interesting combination of point clouds and wave signals to reconstruct the geometry of internal defects of as-built structures in concrete additive manufacturing.

Choosing a geometric representation is often about trade-offs – representations beyond the ubiquitous triangle mesh can make certain tasks fundamentally easier, but come with new challenges. In this talk, I will give two examples of these trade-offs I have encountered in my research by discussing some issues that arise when computing geometric intersections on higher-order patches and neural SDFs. I’ll then discuss some of my work that builds towards solving these issues: detecting intersections between higher-order patches using an optimization technique called sum-of-squares relaxation and computing constructive solid geometry operations on neural SDFs.

We will have a closer look at three surface approximation techniques: the alternating point distance minimisation (A-PDM), an alternating method that is a hybrid between the PDM and the tangent-distance minimization (A-HDM) and the joint optimization method. A common feature of these three methods is that they approximate a parametrised point cloud by not only searching for suitable control points but also by adapting the parameters of the data points. While the literature provides considerable experience with these methods in the curve case, relatively little is known about their performance when constructing surfaces. We have successfully combined the three methods with the truncated hierarchical B-splines (THB-splines) and our examples demonstrate a significant reduction of degrees of freedom necessary to construct a satisfying approximation.

The second focus of the talk will be the industrial viewpoint. Introducing MTU and its activities in developing, manufacturing and repairing aircraft engines will enable putting the methods into an application context. We will review our team’s past experience in industrialising geometric methods; based on this, we will discuss the “boring” and technical part of what the next steps and hurdles towards productive use would be. While the problematics of the geometric modeling at MTU is fairly specific, the principles discussed could be interesting for anyone intent on seeing their methods applied in practice.

Computing the medial axis of a 3D surface mesh is challenging. Points on the discrete medial axis can be defined as interior Voronoï vertices of the surface mesh, but the resulting medial structure rarely has clean connectivity and consistent geometry. In this work, we propose a medial axis computation of an arbitrary surface mesh based on the Voronoï diagram able to generate manifold medial sheets with coherent topology in terms of homothopy and orientability, that is, generating consistent geometric structures similar to those in the continuous setting. Because of the correspondences between the surface mesh and resulting medial mesh, we also provide an efficient method for separating the shape into coherent regions associated to medial structures. This correspondence allows for a medial-axis-based filtration of surface structures to generate a built-in Hausdorff $ϵ$-approximation of the surface points based on a simplified medial axis, thereby providing a robust medial representation with guaranteed surface approximation.

Recent developments in machine learning from sensor data have enabled new possibilities in geometry reconstruction. We will discuss some of these new methods, such as neural implict representations and autoregressive models, and investigate some preliminary results.

Advances in manufacturing processes and design exploration technology have enabled creation of objects with unprecedented complexity and customizability. Design exploration methods such as multi-disciplinary optimization, topology optimization, and AI have enabled rapid exploration of large design spaces. Additive manufacturing enables production of shapes with complex topology and geometric features varying across many length scales. The choice of a geometric representation, and the core modeling algorithms it enables, are crucial in order for a design system to enable engineers to leverage the benefits of these advances and create better mechanical products. Most, if not all, major CAD software systems today are built on Boundary Representations (B-Reps). In our view B-Reps have reached their limits on addressing the design opportunities available today. At nTop, we are building a design system using hybrid implicit modeling that presents advantages in terms of geometric scalability, modeling operation reliability that enables automation, and performance over B-Reps. Our system also provides gradients that are beneficial for design optimization. Using a different geometric representation introduces challenges with interoperability with existing B-Rep based systems. We will present recent advances in this area in creating B-Reps from implicits as well as in direct interop with implicits with an API for our kernel.

This work proposes two, geodesic-curvature based, criteria for shape-preserving interpolation on smooth surfaces, the first criterion being of non-local nature, while the second criterion is a local (weaker) version of the first one. These criteria are tested against a family of on-surface $C^2$ splines obtained by composing the parametric representation of the supporting surface with variabledegree ($\ge$ 3) splines amended with the preimages of the shortest-path geodesic arcs connecting each pair of consecutive interpolation points. After securing that the interpolation problem is well posed, we proceed to investigate the asymptotic behaviour of the proposed on-surface splines as degrees increase. Firstly, it is shown that the local-convexity sub-criterion of the local criterion is satisfied. Second, moving to non-local asymptotics, we prove that, as degrees increase, the interpolant tends uniformly to the spline curve consisting of the shortest-path geodesic arcs. Then, focusing on isometrically parametrized developable surfaces, sufficient conditions are derived, which secure that all criteria of the first (strong) criterion for shape-preserving interpolation are met. Finally, it is proved that, for adequately large degrees, the aforementioned sufficient conditions are satisfied. This permits to build an algorithm that, after a finite number of iterations, provides a $C^2$ shape-preserving interpolant for a given data set on a developable surface.

The numerical solution of partial differential equations (PDE) is ubiquitously used for physical simulation in scientific computing, computer graphics, and engineering. Ideally, a PDE solver should be opaque: the user provides as input the domain boundary, boundary conditions, and the governing equations, and the code returns an evaluator that can compute the value of the solution at any point of the input domain. This is surprisingly far from being the case for all existing open-source or commercial software, despite the research efforts in this direction and the large academic and industrial interest. To a large extent, this is due to lack of robustness and generality in the geometry processing algorithms used to convert raw geometrical data into a format suitable for a PDE solver.

I will discuss the limitations of the current state of the art, and present a proposal for an integrated pipeline, considering data acquisition, meshing, basis design, and numerical optimization as a single challenge, where tradeoffs can be made between different phases to increase automation and efficiency. I will demonstrate that this integrated approach offers many advantages, while opening exciting new geometry processing challenges, and that a fully opaque meshing and analysis solution is already possible for heat transfer and elasticity problems with contact. I will present a set of applications enabled by this approach in reinforcement learning for robotics, force measurements in biology, shape design in mechanical engineering, stress estimation in biomechanics, and simulation of deformable objects in graphics.

Polyhedral-net splines generalize tensor-product splines by allowing additional control net configurations: isotropic patterns, like n quads surrounding a vertex, an n-gon surrounded by quads, and preferred direction patterns, that adjust parameter line density, such as T-junctions. PnS2 generalize C1 bi-2 splines. PnS3 generalize C2 bi-3 splines.

There are two instances of PnS2 in the public domain: a Blender add-on and a ToMS distribution. A web interface now offers solving elliptic PDEs on PnS2 surfaces using PnS2 finite elements. The output for PnS2 is tensor-product polynomial pieces in BB-form (Bezier form).

Mixed integer optimization (MIO) is leveraged regularly across aerospace engineering, addressing problems in the design, manufacturing, production, and in-service life of commercial aircraft. This presentation will start with a brief overview of how Boeing’s Applied Math Group applies MIO to problems at Boeing. This presentation will then focus on a mixed integer nonlinear programming (MINLP) formulation for stringer centerline design in a commercial vehicle. Fuselage stringers are load-bearing components that run lengthwise along an aircraft and transfer aerodynamic loads acting on the skin into the stringers and frames. The design of stringer centerlines in commercial airplanes has traditionally been performed manually by structural engineers since stringer configurations are subject to a wide variety of design and integration requirements. These requirements include minimum and maximum spacing between pairs of centerlines, maximum area constraints on frame bays, and a host of integration requirements with additional features in the fuselage. The MINLP formulation uses discrete variables to assign stringers to drop out of the design at specific frame stations and continuous variables to control the path stringers take on the fuselage surface. We will efficiently enumerate this design space by checking the feasibility of several linear programs representing relaxations of the stringer spacing constraints. This enumeration is incorporated into a branch and bound algorithm that solves the centerline design problem to global optimality. This algorithm is compared to a piecewise linear modeling approach to solve the MINLP.

Additive manufacturing holds enormous potential as a fabrication methodology, offering “free complexity” and flexible, sustainable, local manufacturing. Examples of lightweight parts, high-efficiency heat exchangers, mass customized medical devices and the explosion of new results in metamaterials illustrate the opportunities. Yet, the growth of the AM industry has been slower than projected. Estimates of industry growth from 2020 showed a 25-30% year over year growth, but were recently adjusted downward to a new projected growth of just 13.9% in 2024 (source: AM Power). Some of the barriers to growth include cost (which remains high), a labour shortage, and a general conservatism/change resistance from the manufacturing industry. Key technical barriers to adoption include repeatability/consistency, material characterization, part qualification, availability and education on DfAM tools, and interoperability. Three observations from the perspective of my company Metafold are that:

1. 1.

   manufacturing is increasingly defined by software,

2. 2.

   iteration speed is under pressure for commercial manufacturing, putting pressure on simulation-informed design, and

3. 3.

   appetite for trusted simulation is enormous.

Underpinning these themes is the need for a robust digital foundation. Choosing an appropriate geometry representation is critical to ensuring a seamless workflow between design, simulation, production, and validation, yet available software is divided in approach between meshes, BReps, and implicits, making interoperability a challenge.

The past decade has seen a large variety of new multi-sided surface representations. These works focused primarily on the patch equation, while the input – boundary constraints or control points – was assumed to be given. In our recent publications we have investigated methods for the automatic creation of natural control structures, and we have also proposed different tools for the intuitive modification of multi-sided, multi-connected surfaces.

For the placement of control points we recommend an algorithm based on the (curved) domain, which is in turn generated from the boundary curves. A refinement routine resembling degree elevation provides a means for adding further details to the patch, and also ensures a reasonable distribution of controls.

In the context of editing, control points can be divided into two types: (1) boundary control points (BCPs), responsible for the continuous connection to adjacent surfaces, and (2) interior control points (ICPs), governing the middle of the patch. BCPs have associated algebraic or numerical constraints, rendering direct modification infeasible. Here we propose an indirect approach for modification via control vectors. On the other hand, ICPs are unconstrained, but there are generally too many of them to handle manually. We show how hierarchical and proportional editing schemes can be useful in this situation.

Splines over triangulations or tetrahedral partitions are useful in many applications, such as finite element analysis (FEA), computer aided design (CAD), and other engineering problems. For several of these applications, $C^0$ piecewise linear polynomials do not suffice. In some cases, one needs smoother elements for modeling, or higher polynomial degrees to increase the approximation order. The smoothness over a triangular partition is obtained either by raising the degree of polynomials or keeping the degrees low and splitting the triangles into subtriangles. In the context of Isogeometric Analysis (IgA), the key concept is the development of a new isoparametric paradigm for FEA, where the same basis functions used for geometry representations in CAD systems are adopted for the approximation of field variables. In its original formulation, IgA is based on tensor-product B-splines and their rational version NURBS. Unfortunately, the tensor-product structure has some drawbacks, especially when the adaptivity of the mesh is required. This motivates the interest in alternative structures for IgA, including T-splines, hierarchical B-splines, LR-splines, THB-splines. However, due to their (local) tensor-product structures, there are still restrictions on the refinement. To overcome the tensor-product restriction, many advances have been made in using non-tensor product splines, such as triangular/tetrahedral Bezier patches, Powell-Sabin splines, Box-splines, and so forth. An alternative to the above so-called macro-elements are spline spaces spanned by compactly supported, smooth functions. A well-understood example of such functions is the so-called simplex spline. Recently in[1] it was proposed a new simplex-spline basis for spline spaces built on Clough-Tocher splits with $C^1$ smoothness defined on a general triangulation. In this talk we present the use of spline spaces based on this particular class of simplex splines as a tool in numerical simulation.

Mass lumping techniques are commonly employed in explicit time integration schemes for problems in structural dynamics to both avoid solving costly linear systems with the consistent mass matrix and increase the critical time step. In isogeometric analysis, the critical time step is constrained by so-called “outlier” frequencies, representing the inaccurate high frequency part of the spectrum. Removing or dampening these high frequencies is paramount for fast explicit solution techniques. In this work, we propose robust mass lumping and outlier removal techniques for nontrivial geometries, including multipatch and trimmed geometries. Our lumping strategies provably do not deteriorate (and often improve) the CFL condition of the original problem and are combined with deflation techniques to remove persistent outlier frequencies. Numerical experiments reveal the advantages of the method, especially for simulations covering large time spans where they may halve the number of iterations with little or no effect on the numerical solution.

This talk considers a generalization of the Clough-Tocher interpolant to star-shaped polygons. We show how to automatically tessellate the shape into triangle elements. These elements are quartic functions that have cubic boundaries and quadratic cross-boundary derivatives. We then show how to satisfy all of the smoothness constraints including an exact count of the numbers of degrees of freedom. We remove those degrees of freedom through a fairing functional that reproduces cubic functions and show pictures of several basis functions created via this approach.

In isogeometric analysis, isogeometric function spaces are used for accurately representing the solution to a partial differential equation (PDE) on a parameterized domain. They are generated from a tensor-product spline space by composing its basis functions with the inverse of a parameterization of the physical domain. Depending on the geometry of the domain and on the data of the PDE, the solution might not have maximum Sobolev regularity close to every point of the domain, leading to a reduced convergence rate. In this case, it is necessary to reduce the local mesh size close to the singularities. Based on the concept of $r$-adaptivity we can find a suitable isogeometric function space for a given PDE without sacrificing the tensor-product structure. In particular, we use the fact that different reparameterizations of the same computational domain lead to different isogeometric function spaces while preserving the geometry. Starting from a multi-patch domain consisting of bilinearly parameterized patches, we aim to find the biquadratic multi-patch parameterization of the domain that leads to the smallest approximation error of the solution. In order to estimate the location of the optimal control points, we use a trained residual neural network that predicts optimal parameters for point sets sampled from the graph surface of an approximate solution to the PDE. An iterative procedure leads to a reparameterization of the computational domain that is adapted to the solution of the given PDE and leads to vastly improved approximation errors.

In the evolving landscape of additive manufacturing (AM), the integration of parametric geometric modeling with cloud-based platforms is revolutionizing single-lot production and mass customization. This talk delves into the commercial and technological advancements enabled by our parametric modeling techniques, which are pivotal in harnessing the full potential of AM. We present compelling industry and medical use cases that demonstrate significant enhancements in design efficiency and customization, achieved through our innovative algorithms. Our discussion includes a detailed examination of a novel mesh morphing strategy, which innovatively employs field lines of an (1/r) potential to seamlessly enfold one mesh onto another. This method streamlines the design process and ensures precision and adaptability in complex geometries.

Smooth splines of low degree on triangular or simplicial partitions are considered. These kind of spaces have applications in computer aided design and isogeometric analysis. One way to obtain both smoothness and low degree is to split each element in the space into sub-pieces. Examples we consider are splits named after Clough-Tocher, Alfeld, Powell-Sabin and Wang-She.

On a split we want to construct a basis of multivariate B-splines known as simplex splines and then use Bernstein-Bézier techniques to obtain a global representation.

The construction of spline spaces for the isogeometric analysis of higher order PDEs is partially understood for bivariate problems, but offers great challenges for trivariate problems. In this talk, we discuss the current situation and identify tasks to be addressed in the future. In particular, we consider volumetric subdivision as a possible candidate for the construction of function spaces with sufficient Sobolev regularity. While many analytic questions still remain unsolved, we can report on progress concerning the construction of algorithms with favorable properties.

Generative manufacturing applies the power of artificial intelligence (AI) to generate and execute optimal solutions given customer-defined constraints and parameters, such as functional specifications, cost, and lead time, by exploring vast combinations of design and production alternatives based on material and process availability. In this talk, we present our latest research on combining AI with isogeometric analysis (IGA) for applications in additive manufacturing (AM). It includes a machine learning (ML) framework for inverse design and manufacturing of self-assembling fiber-reinforced composites in 4D printing [1], IGA-based topology optimization for AM of heat exchangers [2, 3], as well as data-driven residual deformation prediction to enhance metal component printability and lattice support structure design in the laser powder bed fusion (LPBF) AM process [4]. By speeding up geometry distortion predictions from several hours to mere seconds with uncertainty quantification, our model can be deployed to prevent generation of infeasible designs. Our on-going efforts also include developing digital twins to enable prediction and control of process parameters for minimal melt pool variability in LPBF manufacturing, where reduced order modeling [5, 6] is one key technique to efficiently simulate underlying physics such as the melt pool dynamics.

Objects made from different materials have demonstrated their potential to offer distinct functional properties across regions. They can be customized to meet specific application requirements by properly allocating materials to areas in need, showcasing remarkable design flexibility. However, multi-material 3D printing technology for fabricating such objects is usually limited by the number of printable base materials. This talk begins with a framework for design, optimization and fabrication of deformable 3D objects and then introduces an approach for efficiently designing the distribution of available base materials within an object to achieve the desired deformation behavior. The approach takes displacements and forces at a set of mesh vertices as input and uses FEM to compute the material distribution. Two formulations are proposed. The first one is a discrete formulation based on L0-minimization, achieving the computation of sparse material distribution in one step, which is beneficial for additive manufacturing with multi-material printers. The second one is a continuous formulation through mathematical relaxation, which facilitates the numerical process. The work is partially supported by MOE AcRF Tier 1 Grant of Singapore (RG12/22).