Digital twin can be defined as a virtual representation of a physical asset enabled through data and simulators for real-time prediction, optimization, monitoring, controlling, and improved decision making. Recent advances in computational pipelines, multiphysics solvers, artificial intelligence, big data cybernetics, data processing and management tools bring the promise of digital twins and their impact on society closer to reality. Digital twinning is now an important and emerging trend in many applications. Also referred to as a computational megamodel, device shadow, mirrored system, avatar or a synchronized virtual prototype, there can be no doubt that a digital twin plays a transformative role not only in how we design and operate cyber-physical intelligent systems, but also in how we advance the modularity of multi-disciplinary systems to tackle fundamental barriers not addressed by the current, evolutionary modeling practices. In this work, we review the recent status of methodologies and techniques related to the construction of digital twins mostly from a modeling perspective. Our aim is to provide a detailed coverage of the current challenges and enabling technologies along with recommendations and reflections for various stakeholders.

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Digital Twin: Values, Challenges and Enablers From a Modeling Perspective

We will deal with the translation surfaces which are the shapes generated by translating one curve along another one. We focus on the geometry of translation surfaces generated by two algebraic curves in space and study their properties, especially those useful for geometric modelling purposes. It is a classical result that each minimal surface may be obtained as a translation surface generated by an isotropic curve and its complex conjugate. Thus, we can study the minimal surfaces as special instances of translation surfaces. All the results about translation surfaces will be directly applied also to minimal surfaces. Finally, we present a construction of rational isotropic curves with a prescribed tangent field which leads to the description of all rational minimal surfaces. A close relation to surfaces with Pythagorean normals will be also discussed.

Mathematical Methods for Curves and Surfaces

This review paper collects several results that form part of the theoretical foundation of isogeometric methods. We analyse variational techniques for the numerical resolution of PDEs based on splines or NURBS and we provide optimal approximation and error estimates in several cases of interest. The theory presented also includes estimates for T-splines, which are an extension of splines allowing for local refinement. In particular, we focus our attention on elliptic and saddle point problems, and we define spline edge and face elements. Our theoretical results are demonstrated by a rich set of numerical examples. Finally, we discuss implementation and efficiency together with preconditioning issues for the final linear system. © Cambridge University Press 2014.

Mathematical analysis of variational isogeometric methods

The finite cell method is an embedded domain method, which combines the fictitious domain approach with higher-order finite elements, adaptive integration, and weak enforcement of unfitted essential boundary conditions. Its core idea is to use a simple unfitted structured mesh of higher-order basis functions for the approximation of the solution fields, while the geometry is captured by means of adaptive quadrature points. This eliminates the need for boundary conforming meshes that require time-consuming and error-prone mesh generation procedures, and opens the door for a seamless integration of very complex geometric models into finite element analysis. At the same time, the finite cell method achieves full accuracy, i.e. optimal rates of convergence, when the mesh is refined, and exponential rates of convergence, when the polynomial degree is increased. Due to the flexibility of the quadrature based geometry approximation, the finite cell method can operate with almost any geometric model, ranging from boundary representations in computer aided geometric design to voxel representations obtained from medical imaging technologies. In this review article, we first provide a concise introduction to the basics of the finite cell method. We then summarize recent developments of the technology, with particular emphasis on the research topics in which we have been actively involved. These include the finite cell method with B-spline and NURBS basis functions, the treatment of geometric nonlinearities for large deformation analysis, the weak enforcement of boundary and coupling conditions, and local refinement schemes. We illustrate the capabilities and advantages of the finite cell method with several challenging examples, e.g. the image-based analysis of foam-like structures, the patient-specific analysis of a human femur bone, the analysis of volumetric structures based on CAD boundary representations, and the isogeometric treatment of trimmed NURBS surfaces. We conclude our review by briefly discussing some key aspects for the efficient implementation of the finite cell method.

The Finite Cell Method: A Review in the Context of Higher-Order Structural Analysis of CAD and Image-Based Geometric Models

Isogeometric analysis using LR B-splines

Polynomial splines over locally refined box-partitions

The use of trivariate NURBS in isogeometric analysis has put the quality of parametrization of NURBS volumes on the agenda Sometimes a NURBS volume needs a better parametrization to meet requirements regarding smoothness, approximation or periodicity In this paper we generalize various smoothing methods that already exist for bivariate parametric spline surfaces to trivariate parametric spline volumes We will also address how rational and polynomial spline volumes create different challenges and solutions in the algorithms

Spline volume fairing

We address progressive local refinement of splines defined on axes parallel box-partitions and corresponding box-meshes in any space dimension. The refinement is specified by a sequence of mesh-rectangles (axes parallel hyperrectangles) in the mesh defining the spline spaces. In the 2-variate case a mesh-rectangle is a knotline segment. When starting from a tensor-mesh this refinement process builds what we denote an LR-mesh, a special instance of a box-mesh. On the LR-mesh we obtain a collection of hierarchically scaled B-splines, denoted LR B-splines, that forms a nonnegative partition of unity and spans the complete piecewise polynomial space on the mesh when the mesh construction follows certain simple rules. The dimensionality of the spline space can be determined using recent dimension formulas [9, 10]. Math Subject Classification: 65D07, 65D17

Locally Refinable Splines over Box-Partitions

The invention provides a method for spatially refining a computer generated l-dimensional (l>0) model in a computing environment, the l-dimensional model representing physical or geometrical properties, and where the l-dimensional model is represented by tensor product B-splines basis functions and l-dimensional coefficients, where the l-dimensional coefficients are in real or projective space, and the tensor product B-splines basis functions are spanning an r-variate spline space (r>0) having a parameter domain, the method comprising: a) inserting at least one axis parallel hyper rectangle degenerate in one dimension in said parameter domain, providing a splitting of a support of at least one of said tensor product B-spline basis functions; b) computing refined tensor product B-spline basis functions by subdivision on said at least one tensor product B-splines basis functions whose support is split, using at least one knot value of the at least one axis parallel hyper rectangle; and c) computing the resulting refined l-dimensional representation based on said refined tensor product B-spline basis functions.

Method for local refinement of a geometric or physical representation

Advances in machine learning (ML) open up possibilities for better supporting the decision making that occurs in high-stakes domains such as air traffic management (ATM). The success of such decision-making systems highly depends upon end users’ involvement in their development process. However, most designers face challenges with finding appropriate ways of doing this. This paper presents our ongoing work to investigate design practices by reporting lessons learned from user involvement in the development of an ML-infused ATM decision support system. To explore if and how UX design methods need to be refined when working with ML as a design material, we conducted an online study with domain experts consisting of three iterations. The paper reports the main challenges we faced and our actions to overcome them. Our results can be useful to other designers working with ML-infused systems.

https://dl.acm.org/doi/pdf/10.1145/3489410.3489421

Kjell Fredrik Pettersen

Papers

Polynomial splines over locally refined box-partitions

Spline volume fairing

Locally Refinable Splines over Box-Partitions

Method for local refinement of a geometric or physical representation

User Involvement in the Design of ML-Infused Systems