Discrete differential operators on polygonal meshes

Geometry processing of surface meshes relies heavily on the discretization of differential operators such as gradient, Laplacian, and covariant derivative. While a variety of discrete operators over triangulated meshes have been developed and used for decades, a similar construction over polygonal meshes remains far less explored despite the prevalence of non-simplicial surfaces in geometric design and engineering applications. This paper introduces a principled construction of discrete differential operators on surface meshes formed by (possibly non-flat and non-convex) polygonal faces. Our approach is based on a novel mimetic discretization of the gradient operator that is linear-precise on arbitrary polygons. Equipped with this discrete gradient, we draw upon ideas from the Virtual Element Method in order to derive a series of discrete operators commonly used in graphics that are now valid over polygonal surfaces. We demonstrate the accuracy and robustness of our resulting operators through various numerical examples, before incorporating them into existing geometry processing algorithms

Morphological Effects of Lignocellulosic Fibres on Poly(Lactic Acid) Biocomposites

Kasvavan ympäristötietoisuuden ja EU:n lainsäädännön myötä biopohjaiset materiaaliratkaisut kiinnostavat entistä enemmän. Lignoselluloosapohjaiset kuitulujitetut biokomposiitit voivat tarjota ympäristöystävällisen materiaalivaihtoehdon monille erilaisille muovituotteille. Biokomposiiteissa fossiilisista lähteistä peräisin olevaa materiaalia korvataan uusiutuvilla raakaainelähteillä, kuten puukuiduilla, joka usein myös laskee tuotteen hiilijalanjälkeä. Puukuitujen käyttö voi parantaa myös materiaalin muita ominaisuuksia, kuten mekaanisia ja haptisia piirteitä. Biokomposiitteja voi myös kierrättää mekaanisesti, mutta taloudellisesti kannattavan kierrätyssysteemin rakentaminen on vaikeaa biokomposiittien toistaiseksi alhaisen volyymin ja jakeiden heterogeenisuuden vuoksi. Sen vuoksi biokomposiitit, jotka pohjautuvat biohajoaviin tai kompostoituviin materiaaleihin, voivat tukea kiertotaloutta tarjoamalla vaihtoehtoisen käsittelytavan tuotteille, jotka eivät sovi kierrätykseen. Toisaalta näiden materiaalien kierrättäminen ei ole poissuljettua. Tässä väitöskirjassa keskitytään puukuitujen morfologian vaikutukseen polylaktidi (PLA) –pohjaisten biokomposiittien ominaisuuksiin. Työn päätavoite oli luoda lisää tietoa näiden materiaalien ominaisuuksiin vaikuttavista tekijöistä, jonka kautta materiaalien laajempi kaupallinen hyödynnettävyys lisääntyy. Tavoitteen saavuttamiseksi tutkimuksessa keskityttiin erityisesti siihen, miten materiaalien sulatyöstö vaikuttaa kuitujen morfologiaan sekä siihen, miten kuidun valinta ja erilaiset kuidun pinnan muokkaukset vaikuttavat ruiskuvalettujen biokomposiittien ominaisuuksiin. Työssä kiinnitettiin erityisesti huomiota kuidun pilkkoontumiseen, dispersioon ja niiden riippuvuuteen komposiitin mekaanisesta suorituskyvystä. Näiden lisäksi työssä tarkasteltiin uusia kevyitä sovellusalueita arvioimalla kuidun lisäyksen vaikutusta PLA:n vaahtoavuuteen ekstruusiovaahdotusprosessissa. Työssä todettiin, että puukuitujen sulatyöstö PLA:n kanssa vaikuttaa väistämättä kuitujen morfologiaan. Kaikki valitut puukuitutyypit lyhenivät samalle tasolle riippumatta kuidun alkuperäisestä pituudesta, kun taas kuitujen halkaisija pysyi muuttumattomana. Näin ollen kuitujen pituus/leveyssuhde oli riippuvainen kuitujen loppupituudesta ja lähtöleveydestä. Kuitujen teollinen valkaisu näytti olevan yksi kuidun tuhoutumiseen vaikuttaneista tekijöistä, sillä valkaistu kuitu on ominaisuuksiltaan helpommin jauhautuvaa. Kuitudispersioiden eroavaisuuden vuoksi suoria johtopäätöksiä kuidun pituus/leveyssuhteen ja komposiitin mekaanisten ominaisuuksien välillä ei voitu kuitenkaan tehdä. Toisaalta kuitujen dispersio ja hienoainepitoisuus näyttivät osittain peilaavan komposiitin mekaanista suorituskykyä. Komposiittien ominaisuudet paranivat käyttämällä lujitteena valkaisematonta sellukuitua, joka säilytti paremmin mittansa sulatyöstön aikana, sekä käyttämällä termomekaanista sellua, fraktioitua kuitua tai dispergointi-/ kompatibilisointiainetta. Pintaligniini ja epoksoitu pellavansiemenöljy näyttivät parantavan myös kuidun ja matriisin välistä yhteensopivuutta. Työssä todettiin myös, että kierrätyskuiduilla, kuten siistaamattomilla sanomalehdillä ja nestepakkauskartongilla, on potentiaalia PLA:n kuitulujitteena. Ekstruusiovaahdotuksessa kuidun lisäys mahdollisti keveiden vaahtojen valmistamisen. Kuitulisäyksellä saavutettiin pienempi solukoko ja suurempi solutiheys kuin puhtaalla PLA:lla. Kaiken kaikkiaan työn yhteenvetona voidaan todeta, että oikealla kuidunvalinnalla ja kuitukäsittelyillä voidaan saavuttaa merkittävä parannus puukuitulujitteisten PLA-biokomposiittien suorituskykyyn.Increasing environmental awareness and tightening EU legislations have gained significant interest towards bio-based material solutions. Lignocellulosic fibre reinforced biocomposites can provide a sustainable material alternative for several plastic products. In biocomposites, renewable raw material sources, such as wood fibres, replace typically fossil-based resources, often decreasing also the carbon footprint of the material. Utilisation of wood fibres also provide additional material improvements to plastics, such as increased mechanical performance and natural haptics. However, while biocomposite materials can be mechanically recycled, building up an economically viable municipal recycling system is challenging due to low material volumes and heterogeneous material streams. Thus, biocomposites based on biodegradable or compostable matrix materials can complement the circular economy by providing an alternative end-of-life solution for products that are not applicable for recycling, but on the other hand, not excluding recyclability. This thesis concentrates on the effects of wood fibre morphologies on biocomposite performance using compostable poly(lactic acid) (PLA) as the matrix material. The main objective was to increase knowledge of the factors influencing the material characteristics to enable their wider use in commercial applications. To reach this objective, the study focused especially to the effect of melt processing on fibre morphologies, as well as the effect of various fibre surface treatments and fibre selection on performance of injection moulded bicomposites. Particular attention was paid to fibre attrition, dispersion and their correlation to mechanical performance. In addition, new light-weight application areas were considered by evaluating the effect of fibre addition on PLA foamability in extrusion foaming process. As a conclusion, it was observed that melt processing of wood fibres with PLA has unpreventable effect on wood fibre morphology. All selected bleached wood fibre types shortened to the same level independent of the initial fibre length while fibre diameters remained unchanged. Thus, aspect ratio depended on retaining fibre length and initial fibre width of the fibres. Industrial fibre bleaching was found to be one factor influencing the increased fibre attrition due to higher refinability of bleached fibres. However, due to differences in fibre dispersion, no straight conclusions could be made of the correlation between fibre aspect ratio after processing and the mechanical properties of the composites. On the other hand, fibre dispersion and fines content seemed to partly reflect the mechanical performance. Improved mechanical properties were obtained by utilisation of unbleached hardwood kraft pulp fibres with low fibre attrition during processing, thermomechanical pulp, fibre fractionation or addition of compatibiliser/dispersing agent to the system. The presence of lignin and epoxidated linseed oil also indicated improvement in fibre-matrix adhesion. The work also presented that recycled fibres, such as non-deinked newspapers and liquid packaging board scratch, have potential to be used as PLA reinforcement. In extrusion foaming process, addition of wood fibres to PLA still enabled the production of low-density foams with decreased cell size and increased cell density. As an overall conclusion, careful selection of fibre type and treatments can provide significantly improved properties for wood fibre reinforced PLA biocomposites

Multiscale modelling of woven and knitted fabric membranes

Light-weight fabric membranes have gained increasing popularity over the past years due to their tailorable structural and material performances. These tailorable properties include stretch forming and deep drawing formability that exhibits excellent stretchability and drapeability properties of textiles and textile composites. Since the inception of computerised numerical control for three-dimensional textile-manufacturing machines, technical textiles paved their way to numerous applications, certainly not limited to; aerospace, biomedical, civil engineering, defence, marine and medical industries. Digital interlooping and digital interlacing technology in additive manufacturing greatly advanced the manufacturing processes of textiles. In this work, we consider two branches of technical fabrics, namely plain-woven and weft-knitted. Multiscale modelling is the tool of choice for homogenising periodic structures and has been used extensively to model and analyse the mechanical behaviour of woven and knitted fabrics. But there is a plethora of literature discussing the demerits of such conventional multiscale modelling. These demerits include higher computational costs, rigid numerical models, ineffcient algorithmic computations and inability to incorporate geometric nonlinearities. We propose a data-driven nonlinear multiscale modelling technique to analyse the complex mechanical behaviour of plain-woven and weft-knitted fabrics with a neat extension to fabric material designing. We show how the integration of statistical learning techniques mitigates the weaknesses of conventional multiscale modelling. Moreover, we discuss the avenues that will open in many potential fields with regard to material modelling, structural engineering and textile industries. In the proposed data-driven nonlinear computational homogenisation technique, we effi ciently integrate the microscale and macroscale using Gaussian Process Regression (GPR) statistical learning technique. In the microscale, representative volume elements (RVEs) are modelled using nite deformable isogeometric spatial rods and deformation is homogenised using periodic boundary conditions. This nite deformable rod is profi cient in handling large deformations, rod-to-rod contacts, arbitrary cross-section de finitions and follower loads. Respecting the principle of separation of scales, we construct response databases by applying different homogenised strain states to the RVEs and recording the respective incremental volume-averaged energy values. We use GPR to learn a model using a 5-fold cross-validation technique by optimising the log marginal likelihood. In the macroscale, textiles are modelled as nonlinear orthotropic membranes for which the stresses and material constitutive relations are predicted by the trained GPR model. This coupling between GPR and membrane models is achieved through a systematic and seamless nite element integration using C++ and Python environments. A neat extension to material designing is also discussed with potentials to extend the work into other related fi elds.Cambridge trust and Trinity Hall scholarshi

Wavelet Theory Demystified

In this paper, we revisit wavelet theory starting from the representation of a scaling function as the convolution of a B-spline (the regular part of it) and a distribution (the irregular or residual part). This formulation leads to some new insights on wavelets and makes it possible to rederive the main results of the classical theory—including some new extensions for fractional orders—in a self-contained, accessible fashion. In particular, we prove that the B-spline component is entirely responsible for five key wavelet properties: order of approximation, reproduction of polynomials, vanishing moments, multiscale differentiation property, and smoothness (regularity) of the basis functions. We also investigate the interaction of wavelets with differential operators giving explicit time domain formulas for the fractional derivatives of the basis functions. This allows us to specify a corresponding dual wavelet basis and helps us understand why the wavelet transform provides a stable characterization of the derivatives of a signal. Additional results include a new peeling theory of smoothness, leading to the extended notion of wavelet differentiability in the $L _{ p }$ -sense and a sharper theorem stating that smoothness implies order

New strategies for curve and arbitrary-topology surface constructions for design

This dissertation presents some novel constructions for curves and surfaces with arbitrary topology in the context of geometric modeling. In particular, it deals mainly with three intimately connected topics that are of interest in both theoretical and applied research: subdivision surfaces, non-uniform local interpolation (in both univariate and bivariate cases), and spaces of generalized splines. Specifically, we describe a strategy for the integration of subdivision surfaces in computer-aided design systems and provide examples to show the effectiveness of its implementation. Moreover, we present a construction of locally supported, non-uniform, piecewise polynomial univariate interpolants of minimum degree with respect to other prescribed design parameters (such as support width, order of continuity and order of approximation). Still in the setting of non-uniform local interpolation, but in the case of surfaces, we devise a novel parameterization strategy that, together with a suitable patching technique, allows us to define composite surfaces that interpolate given arbitrary-topology meshes or curve networks and satisfy both requirements of regularity and aesthetic shape quality usually needed in the CAD modeling framework. Finally, in the context of generalized splines, we propose an approach for the construction of the optimal normalized totally positive (B-spline) basis, acknowledged as the best basis of representation for design purposes, as well as a numerical procedure for checking the existence of such a basis in a given generalized spline space. All the constructions presented here have been devised keeping in mind also the importance of application and implementation, and of the related requirements that numerical procedures must satisfy, in particular in the CAD context

Proceedings of the 35th WIC Symposium on Information Theory in the Benelux and the 4th joint WIC/IEEE Symposium on Information Theory and Signal Processing in the Benelux, Eindhoven, the Netherlands May 12-13, 2014

Compressive sensing (CS) as an approach for data acquisition has recently received much attention. In CS, the signal recovery problem from the observed data requires the solution of a sparse vector from an underdetermined system of equations. The underlying sparse signal recovery problem is quite general with many applications and is the focus of this talk. The main emphasis will be on Bayesian approaches for sparse signal recovery. We will examine sparse priors such as the super-Gaussian and student-t priors and appropriate MAP estimation methods. In particular, re-weighted l2 and re-weighted l1 methods developed to solve the optimization problem will be discussed. The talk will also examine a hierarchical Bayesian framework and then study in detail an empirical Bayesian method, the Sparse Bayesian Learning (SBL) method. If time permits, we will also discuss Bayesian methods for sparse recovery problems with structure; Intra-vector correlation in the context of the block sparse model and inter-vector correlation in the context of the multiple measurement vector problem

Proceedings of the 35th WIC Symposium on Information Theory in the Benelux and the 4th joint WIC/IEEE Symposium on Information Theory and Signal Processing in the Benelux, Eindhoven, the Netherlands May 12-13, 2014

Compressive sensing (CS) as an approach for data acquisition has recently received much attention. In CS, the signal recovery problem from the observed data requires the solution of a sparse vector from an underdetermined system of equations. The underlying sparse signal recovery problem is quite general with many applications and is the focus of this talk. The main emphasis will be on Bayesian approaches for sparse signal recovery. We will examine sparse priors such as the super-Gaussian and student-t priors and appropriate MAP estimation methods. In particular, re-weighted l2 and re-weighted l1 methods developed to solve the optimization problem will be discussed. The talk will also examine a hierarchical Bayesian framework and then study in detail an empirical Bayesian method, the Sparse Bayesian Learning (SBL) method. If time permits, we will also discuss Bayesian methods for sparse recovery problems with structure; Intra-vector correlation in the context of the block sparse model and inter-vector correlation in the context of the multiple measurement vector problem

Optimal Surface Fitting of Point Clouds Using Local Refinement : Application to GIS Data

This open access book provides insights into the novel Locally Refined B-spline (LR B-spline) surface format, which is suited for representing terrain and seabed data in a compact way. It provides an alternative to the well know raster and triangulated surface representations. An LR B-spline surface has an overall smooth behavior and allows the modeling of local details with only a limited growth in data volume. In regions where many data points belong to the same smooth area, LR B-splines allow a very lean representation of the shape by locally adapting the resolution of the spline space to the size and local shape variations of the region. The iterative method can be modified to improve the accuracy in particular domains of a point cloud. The use of statistical information criterion can help determining the optimal threshold, the number of iterations to perform as well as some parameters of the underlying mathematical functions (degree of the splines, parameter representation). The resulting surfaces are well suited for analysis and computing secondary information such as contour curves and minimum and maximum points. Also deformation analysis are potential applications of fitting point clouds with LR B-splines