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Material failure can be tackled by so-called nonlocal models, which introduce an intrinsic length scale into the formulation and, in the case of material failure, restore the well-posedness of the underlying boundary value problem or initial boundary value problem. Among nonlocal models, peridynamics (PD) has attracted a lot of attention as it allows the natural transition from continuum to discontinue and thus allows modeling of discrete cracks without the need to describe and track the crack topology, which has been a major obstacle in traditional discrete crack approaches. This is achieved by replacing the divergence of the Cauchy stress tensor through an integral over so-called bond forces, which account for the interaction of particles. A quasi-continuum approach is then used to calibrate the material parameters of the bond forces, i.e., equating the PD energy with the energy of a continuum. One major issue for the application of PD to general complex problems is that they are limited to fairly simple material behavior and pure mechanical problems based on explicit time integration. PD has been extended to other applications but losing simultaneously its simplicity and ease in modeling material failure. Furthermore, conventional PD suffers from instability and hourglass modes that require stabilization. It also requires the use of constant horizon sizes, which drastically reduces its computational efficiency. The latter issue was resolved by the so-called dual-horizon peridynamics (DH-PD) formulation and the introduction of the duality of horizons.
Within the nonlocal operator method (NOM), the concept of nonlocality is further extended and can be considered a generalization of DH-PD. Combined with the energy functionals of various physical models, the nonlocal forms based on the dual-support concept can be derived. In addition, the variation of the energy functional allows implicit formulations of the nonlocal theory. While traditional integral equations are formulated in an integral domain, the dual-support approaches are based on dual integral domains. One prominent feature of NOM is its compatibility with variational and weighted residual methods. The NOM yields a direct numerical implementation based on the weighted residual method for many physical problems without the need for shape functions. Only the definition of the energy or boundary value problem is needed to drastically facilitate the implementation. The nonlocal operator plays an equivalent role to the derivatives of the shape functions in meshless methods and finite element methods (FEM). Based on the variational principle, the residual and the tangent stiffness matrix can be obtained with ease by a series of matrix multiplications. In addition, NOM can be used to derive many nonlocal models in strong form.
The principal contributions of this dissertation are the implementation and application of NOM, and also the development of approaches for dealing with fractures within the NOM, mostly for dynamic fractures. The primary coverage and results of the dissertation are as follows:
-The first/higher-order implicit NOM and explicit NOM, including a detailed description of the implementation, are presented. The NOM is based on so-called support, dual-support, nonlocal operators, and an operate energy functional ensuring stability. The nonlocal operator is a generalization of the conventional differential operators. Combining with the method of weighted residuals and variational principles, NOM establishes the residual and tangent stiffness matrix of operate energy functional through some simple matrix without the need of shape functions as in other classical computational methods such as FEM. NOM only requires the definition of the energy drastically simplifying its implementation. For the sake of conciseness, the implementation in this chapter is focused on linear elastic solids only, though the NOM can handle more complex nonlinear problems. An explicit nonlocal operator method for the dynamic analysis of elasticity solid problems is also presented. The explicit NOM avoids the calculation of the tangent stiffness matrix as in the implicit NOM model. The explicit scheme comprises the Verlet-velocity algorithm. The NOM can be very flexible and efficient for solving partial differential equations (PDEs). It's also quite easy for readers to use the NOM and extend it to solve other complicated physical phenomena described by one or a set of PDEs. Several numerical examples are presented to show the capabilities of this method.
-A nonlocal operator method for the dynamic analysis of (thin) Kirchhoff plates is proposed. The nonlocal Hessian operator is derived from a second-order Taylor series expansion. NOM is higher-order continuous, which is exploited for thin plate analysis that requires $C^1$ continuity. The nonlocal dynamic governing formulation and operator energy functional for Kirchhoff plates are derived from a variational principle. The Verlet-velocity algorithm is used for time discretization. After confirming the accuracy of the nonlocal Hessian operator, several numerical examples are simulated by the nonlocal dynamic Kirchhoff plate formulation.
-A nonlocal fracture modeling is developed and applied to the simulation of quasi-static and dynamic fractures using the NOM. The phase field's nonlocal weak and associated strong forms are derived from a variational principle. The NOM requires only the definition of energy. We present both a nonlocal implicit phase field model and a nonlocal explicit phase field model for fracture; the first approach is better suited for quasi-static fracture problems, while the key application of the latter one is dynamic fracture. To demonstrate the performance of the underlying approach, several benchmark examples for quasi-static and dynamic fracture are solved.
Renewable energy use is on the rise and these alternative resources of energy can help combat with the climate change. Around 80% of the world's electricity comes from coal and petroleum however, the renewables are the fastest growing source of energy in the world. Solar, wind, hydro, geothermal and biogas are the most common forms of renewable energy. Among them, wind energy is emerging as a reliable and large-scaled source of power production. The recent research and confidence in the performance has led to the construction of more and bigger wind turbines around the world. As wind turbines are getting bigger, a concern regarding their safety is also in discussion. Wind turbines are expensive machinery to construct and the enormous capital investment is one of the main reasons, why many countries are unable to adopt to the wind energy. Generally, a reliable wind turbine will result in better performance and assist in minimizing the cost of operation. If a wind turbine fails, it's a loss of investment and can be harmful for the surrounding habitat. This thesis aims towards estimating the reliability of an offshore wind turbine. A model of Jacket type offshore wind turbine is prepared by using finite element software package ABAQUS and is compared with the structural failure criteria of the wind turbine tower. UQLab, which is a general uncertainty quantification framework developed at ETH Zürich, is used for the reliability analysis. Several probabilistic methods are included in the framework of UQLab, which include Monte Carlo, First Order Reliability Analysis and Adaptive Kriging Monte Carlo simulation. This reliability study is performed only for the structural failure of the wind turbine but it can be extended to many other forms of failures e.g. reliability for power production, or reliability for different component failures etc. It's a useful tool that can be utilized to estimate the reliability of future wind turbines, that could result in more safer and better performance of wind turbines.
Finite Element Simulations of dynamically excited structures are mainly influenced by the mass, stiffness, and damping properties of the system, as well as external loads. The prediction quality of dynamic simulations of vibration-sensitive components depends significantly on the use of appropriate damping models. Damping phenomena have a decisive influence on the vibration amplitude and the frequencies of the vibrating structure. However, developing realistic damping models is challenging due to the multiple sources that cause energy dissipation, such as material damping, different types of friction, or various interactions with the environment.
This thesis focuses on thermoelastic damping, which is the main cause of material damping in homogeneous materials. The effect is caused by temperature changes due to mechanical strains. In vibrating structures, temperature gradients arise in adjacent tension and compression areas. Depending on the vibration frequency, they result in heat flows, leading to increased entropy and the irreversible transformation of mechanical energy into thermal energy.
The central objective of this thesis is the development of efficient simulation methods to incorporate thermoelastic damping in finite element analyses based on modal superposition. The thermoelastic loss factor is derived from the structure's mechanical mode shapes and eigenfrequencies. In subsequent analyses that are performed in the time and frequency domain, it is applied as modal damping.
Two approaches are developed to determine the thermoelastic loss in thin-walled plate structures, as well as three-dimensional solid structures. The realistic representation of the dissipation effects is verified by comparing the simulation results with experimentally determined data. Therefore, an experimental setup is developed to measure material damping, excluding other sources of energy dissipation.
The three-dimensional solid approach is based on the determination of the generated entropy and therefore the generated heat per vibration cycle, which is a measure for thermoelastic loss in relation to the total strain energy. For thin plate structures, the amount of bending energy in a modal deformation is calculated and summarized in the so-called Modal Bending Factor (MBF). The highest amount of thermoelastic loss occurs in the state of pure bending. Therefore, the MBF enables a quantitative classification of the mode shapes concerning the thermoelastic damping potential.
The results of the developed simulations are in good agreement with the experimental results and are appropriate to predict thermoelastic loss factors. Both approaches are based on modal superposition with the advantage of a high computational efficiency. Overall, the modeling of thermoelastic damping represents an important component in a comprehensive damping model, which is necessary to perform realistic simulations of vibration processes.
The aim of this study is controlling of spurious oscillations developing around discontinuous solutions of both linear and non-linear wave equations or hyperbolic partial differential equations (PDEs). The equations include both first-order and second-order (wave) hyperbolic systems. In these systems even smooth initial conditions, or smoothly varying source (load) terms could lead to discontinuous propagating solutions (fronts). For the first order hyperbolic PDEs, the concept of central high resolution schemes is integrated with the multiresolution-based adaptation to capture properly both discontinuous propagating fronts and effects of fine-scale responses on those of larger scales in the multiscale manner. This integration leads to using central high resolution schemes on non-uniform grids; however, such simulation is unstable, as the central schemes are originally developed to work properly on uniform cells/grids. Hence, the main concern is stable collaboration of central schemes and multiresoltion-based cell adapters. Regarding central schemes, the considered approaches are: 1) Second order central and central-upwind schemes; 2) Third order central schemes; 3) Third and fourth order central weighted non-oscillatory schemes (central-WENO or CWENO); 4) Piece-wise parabolic methods (PPMs) obtained with two different local stencils. For these methods, corresponding (nonlinear) stability conditions are studied and modified, as well. Based on these stability conditions several limiters are modified/developed as follows: 1) Several second-order limiters with total variation diminishing (TVD) feature, 2) Second-order uniformly high order accurate non-oscillatory (UNO) limiters, 3) Two third-order nonlinear scaling limiters, 4) Two new limiters for PPMs. Numerical results show that adaptive solvers lead to cost-effective computations (e.g., in some 1-D problems, number of adapted grid points are less than 200 points during simulations, while in the uniform-grid case, to have the same accuracy, using of 2049 points is essential). Also, in some cases, it is confirmed that fine scale responses have considerable effects on higher scales.
In numerical simulation of nonlinear first order hyperbolic systems, the two main concerns are: convergence and uniqueness. The former is important due to developing of the spurious oscillations, the numerical dispersion and the numerical dissipation. Convergence in a numerical solution does not guarantee that it is the physical/real one (the uniqueness feature). Indeed, a nonlinear systems can converge to several numerical results (which mathematically all of them are true). In this work, the convergence and uniqueness are directly studied on non-uniform grids/cells by the concepts of local numerical truncation error and numerical entropy production, respectively. Also, both of these concepts have been used for cell/grid adaptations. So, the performance of these concepts is also compared by the multiresolution-based method. Several 1-D and 2-D numerical examples are examined to confirm the efficiency of the adaptive solver. Examples involve problems with convex and non-convex fluxes. In the latter case, due to developing of complex waves, proper capturing of real answers needs more attention. For this purpose, using of method-adaptation seems to be essential (in parallel to the cell/grid adaptation). This new type of adaptation is also performed in the framework of the multiresolution analysis.
Regarding second order hyperbolic PDEs (mechanical waves), the regularization concept is used to cure artificial (numerical) oscillation effects, especially for high-gradient or discontinuous solutions. There, oscillations are removed by the regularization concept acting as a post-processor. Simulations will be performed directly on the second-order form of wave equations. It should be mentioned that it is possible to rewrite second order wave equations as a system of first-order waves, and then simulated the new system by high resolution schemes. However, this approach ends to increasing of variable numbers (especially for 3D problems).
The numerical discretization is performed by the compact finite difference (FD) formulation with desire feature; e.g., methods with spectral-like or optimized-error properties. These FD methods are developed to handle high frequency waves (such as waves near earthquake sources). The performance of several regularization approaches is studied (both theoretically and numerically); at last, a proper regularization approach controlling the Gibbs phenomenon is recommended.
At the end, some numerical results are provided to confirm efficiency of numerical solvers enhanced by the regularization concept. In this part, shock-like responses due to local and abrupt changing of physical properties, and also stress wave propagation in stochastic-like domains are studied.
Die Qualität von Beplankungselementen wirkt sich deutlich auf den Feuerwiderstand von Metallständer-Wandkonstruktionen aus. Daher wurde im Rahmen dieser Arbeit der Einfluss von Zusätzen in Gipsplatten bezüglich einer möglichen Verbesserung dieser Eigenschaft untersucht.
Zu diesem Zweck wurden spezielle, den jeweiligen Untersuchungsbedingungen angepasste Probekörper unter Verwendung verschiedenster Zusätze gefertigt. Die Beurteilung deren Auswirkungen erfolgte insbesondere mittels nachfolgender fünf Kriterien:
1) dem Zeitpunkt der Temperaturerhöhung nach der Probekörperentwässerung,
2) dem Maximalwert der Plattenrückseitentemperatur,
3) der Größe und der Anzahl der Risse,
4) der Plattenstabilität nach der Wärmebeanspruchung,
5) der Verkürzung von prismatischen Probekörpern.
Besonders wichtig war hierbei die Charakterisierung der Auswirkungen einer simulierten Brandbeanspruchung von 970 °C über 90 Minuten auf Labor-Gipsplatten. Dabei wurde die Temperaturänderung auf der Plattenrückseite über den gesamten Prüfzeitraum kontinuierlich erfasst. Die Bewertung des Zusammenhalts der Platten nach der thermischen Beanspruchung erfolgte erstmals quantitativ über Anzahl und Größe der an den Proben entstandenen Risse. Ursächlich für die Rissbildung ist die Verringerung des Probekörpervolumens infolge des ausgetriebenen Kristallwassers. Da dieser Parameter im Plattenversuch nicht bestimmt werden kann, wurde ergänzend das Längenänderungsverhalten von Prismen im Ergebnis einer 90minütigen Temperung bei 1000 °C im Muffelofen ermittelt.
Besonders vorteilhaft hat sich die Zugabe von 80 g/m2 Glasfasern und 7,75 % Kalksteinmehl auf das Verhalten von Gipsplatten bei Brandbeanspruchung ausgewirkt. Diese Verbesserung ist insbesondere auf höhere Stabilität und geringere Schrumpfung der Gipsplatte zurückzuführen.
Basierend auf den im Labormaßstab erhaltenen Ergebnissen wurden Rezepturvorschläge zur Verbesserung des Feuerwiderstandsverhaltens von Gipsplatten unter Praxisbedingungen entwickelt. Die Herstellung der erforderlichen großformatigen Platten erfolgte auf der Bandstraße der Knauf Gips KG. Diese Platten wurden als Wandkonstruktion mit zweilagiger Beplankung einer großtechnischen Prüfung erfolgreich unterzogen. Eine geringere Durchbiegung der Wandkonstruktion, eine verminderte Volumenreduzierung der Platten sowie eine erhöhte Plattenstabilität belegen die verbesserten Eigenschaften dieser modifizierten Feuerschutzplatte.
Weitere durchgeführte Untersuchungen ergaben, dass es unerheblich ist, ob die Platten auf Basis von Natur- oder REA-Gips bzw. mit hohem oder niedrigem Flächengewicht gefertigt wurden. Das eindeutig beste Ergebnis mit einer Feuerwiderstandsdauer von 118 Minuten hat eine Wandkonstruktion aus Feuerschutzplatten auf Basis eines Stuckgipses aus 100 % REA-Gips mit einem Anteil von 83,9 g/m2 Glasfasern und 1 % Vermiculit und einem Flächengewicht von 10,77 kg/m2, bei einer Plattenstärke von 12,5 mm.
Die als Ziel vorgebende Feuerwiderstandsdauer von 120 Minuten bei zweilagiger Beplankung ohne Dämmstoff könnte künftig erreicht werden, wenn es gelingt, die Volumenreduzierung noch besser zu kompensieren und die Plattenstabilität zu steigern. Eine Möglichkeit hierzu ist die Substitution der beidseitigen Kartonlagen durch eine Glasfaser-Vliesummantelung. Die Wandkonstruktion W112 ohne Dämmstoff erreicht dabei eine Feuerwiderstandsdauer von weit über 120 Minuten, wobei der Gipskern mit Glasfasern armiert ist.
Landwirtschaftliche Biogasanlagen leisten mit ca. 9.300 Anlagen und einem Anteil von 5,3% an der Stromerzeugung, einen Beitrag zur Erzeugung Erneuer-barer Energien in Deutschland. Die Optimierung dieser Anlagen fördert die nachhaltige Bereitstellung von Strom, Wärme und BioErdgas.
Das Ergebnis dieser Forschungsarbeit ist die Entwicklung eines mehrmethodi-schen Bewertungsansatzes zur Beschreibung der Qualität der Eingangs-substrate als Teil einer ganzheitlichen Prozessoptimierung. Dies gelingt durch die kombinierte Nutzung klassischer Analysesätze, der Nutzung organolepti-scher Parameter – der humansensorischen Sinnenprüfung – und der Integration von prozess- und substratspezifischem Erfahrungswissen. Anhand von halbtechnischen Versuchen werden Korrelationen und Kausalitäten zwi-schen chemisch-physikalischen, biologischen, organoleptischen und erfahrungsbezogenen Parametern erforscht. Die Entwicklung einer Fallbasis mit Hilfe des Fallbasierten Schließens, einer Form Künstlicher Intelligenz, zeigt das Entwicklungs- und Integrationspotenzial der Automatisierung auf, insbesondere auch im Hinblick auf neue Ansätze z.B. Industrie 4.0. Erste Lösungen zur Bewältigung der identifizierten Herausforderungen der mehrmethodischen Prozessbewertung werden vorgestellt.
Abschließend wird ein Ausblick auf den weiteren Forschungsbedarf gegeben und die Übertragbarkeit des mehrmethodischen Bewertungsansatzes auf andere Anwendungsfelder z.B. Bioabfallbehandlung, Kläranlagen angeregt.
Das Ziel der Arbeit ist, eine mögliche Verbesserung der Güte der Lebensdauervorhersage für Gusseisenwerkstoffe mit Kugelgraphit zu erreichen, wobei die Gießprozesse verschiedener Hersteller berücksichtigt werden.
Im ersten Schritt wurden Probenkörper aus GJS500 und GJS600 von mehreren Gusslieferanten gegossen und daraus Schwingproben erstellt.
Insgesamt wurden Schwingfestigkeitswerte der einzelnen gegossenen Proben sowie der Proben des Bauteils von verschiedenen Gussherstellern weltweit entweder durch direkte Schwingversuche oder durch eine Sammlung von Betriebsfestigkeitsversuchen bestimmt.
Dank der metallografischen Arbeit und Korrelationsanalyse konnten drei wesentliche Parameter zur Bestimmung der lokalen Dauerfestigkeit festgestellt werden: 1. statische Festigkeit, 2. Ferrit- und Perlitanteil der Mikrostrukturen und 3. Kugelgraphitanzahl pro Flächeneinheit.
Basierend auf diesen Erkenntnissen wurde ein neues Festigkeitsverhältnisdiagramm (sogenanntes Sd/Rm-SG-Diagramm) entwickelt.
Diese neue Methodik sollte vor allem ermöglichen, die Bauteildauerfestigkeit auf der Grundlage der gemessenen oder durch eine Gießsimulation vorhersagten lokalen Zugfestigkeitswerte sowie Mikrogefügenstrukturen besser zu prognostizieren.
Mithilfe der Versuche sowie der Gießsimulation ist es gelungen, unterschiedliche Methoden der Lebensdauervorhersage unter Berücksichtigung der Herstellungsprozesse weiterzuentwickeln.
Scalarization methods are a category of multiobjective optimization (MOO) methods. These methods allow the usage of conventional single objective optimization algorithms, as scalarization methods reformulate the MOO problem into a single objective optimization problem. The scalarization methods analysed within this thesis are the Weighted Sum (WS), the Epsilon-Constraint (EC), and the MinMax (MM) method. After explaining the approach of each method, the WS, EC and MM are applied, a-posteriori, to three different examples: to the Kursawe function; to the ten bar truss, a common benchmark problem in structural optimization; and to the metamodel of an aero engine exit module.
The aim is to evaluate and compare the performance of each scalarization method that is examined within this thesis. The evaluation is conducted using performance metrics, such as the hypervolume and the generational distance, as well as using visual comparison.
The application to the three examples gives insight into the advantages and disadvantages of each method, and provides further understanding of an adequate application of the methods concerning high dimensional optimization problems.
Tropical coral reefs, one of the world’s oldest ecosystems which support some of the highest levels of biodiversity on the planet, are currently facing an unprecedented ecological crisis during this massive human-activity-induced period of extinction. Hence, tropical reefs symbolically stand for the destructive effects of human activities on nature [4], [5]. Artificial reefs are excellent examples of how architectural design can be combined with ecosystem regeneration [6], [7], [8]. However, to work at the interface between the artificial and the complex and temporal nature of natural systems presents a challenge, i.a. in respect to the B-rep modelling legacy of computational modelling.
The presented doctorate investigates strategies on how to apply digital practice to realise what is an essential bulwark to retain reefs in impossibly challenging times. Beyond the main question of integrating computational modelling and high precision monitoring strategies in artificial coral reef design, this doctorate explores techniques, methods, and linking frameworks to support future research and practice in ecology led design contexts.
Considering the many existing approaches for artificial coral reefs design, one finds they often fall short in precisely understanding the relationships between architectural and ecological aspects (e.g. how a surface design and material composition can foster coral larvae settlement, or structural three-dimensionality enhance biodiversity) and lack an integrated underwater (UW) monitoring process. Such a process is necessary in order to gather knowledge about the ecosystem and make it available for design, and to learn whether artificial structures contribute to reef regeneration or rather harm the coral reef ecosystem.
For the research, empirical experimental methods were applied: Algorithmic coral reef design, high precision UW monitoring, computational modelling and simulation, and validated through parallel real-world physical experimentation – two Artificial Reef Prototypes (ARPs) in Gili Trawangan, Indonesia (2012–today). Multiple discrete methods and sub techniques were developed in seventeen computational experiments and applied in a way in which many are cross valid and integrated in an overall framework that is offered as a significant contribution to the field. Other main contributions include the Ecosystem-aware design approach, Key Performance Indicators (KPIs) for coral reef design, algorithmic design and fabrication of Biorock cathodes, new high precision UW monitoring strategies, long-term real-world constructed experiments, new digital analysis methods and two new front-end web-based tools for reef design and monitoring reefs. The methodological framework is a finding of the research that has many technical components that were tested and combined in this way for the very first time.
In summary, the thesis responds to the urgency and relevance in preserving marine species in tropical reefs during this massive extinction period by offering a differentiated approach towards artificial coral reefs – demonstrating the feasibility of digitally designing such ‘living architecture’ according to multiple context and performance parameters. It also provides an in-depth critical discussion of computational design and architecture in the context of ecosystem regeneration and Planetary Thinking. In that respect, the thesis functions as both theoretical and practical background for computational design, ecology and marine conservation – not only to foster the design of artificial coral reefs technically but also to provide essential criteria and techniques for conceiving them.
Keywords: Artificial coral reefs, computational modelling, high precision underwater monitoring, ecology in design.
Isogeometric analysis (IGA) is a numerical method for solving partial differential equations (PDEs), which was introduced with the aim of integrating finite element analysis with computer-aided design systems. The main idea of the method is to use the same spline basis functions which describe the geometry in CAD systems for the approximation of solution fields in the finite element method (FEM). Originally, NURBS which is a standard technology employed in CAD systems was adopted as basis functions in IGA but there were several variants of IGA using other technologies such as T-splines, PHT splines, and subdivision surfaces as basis functions. In general, IGA offers two key advantages over classical FEM: (i) by describing the CAD geometry exactly using smooth, high-order spline functions, the mesh generation process is simplified and the interoperability between CAD and FEM is improved, (ii) IGA can be viewed as a high-order finite element method which offers basis functions with high inter-element continuity and therefore can provide a primal variational formulation of high-order PDEs in a straightforward fashion. The main goal of this thesis is to further advance isogeometric analysis by exploiting these major advantages, namely precise geometric modeling and the use of smooth high-order splines as basis functions, and develop robust computational methods for problems with complex geometry and/or complex multi-physics.
As the first contribution of this thesis, we leverage the precise geometric modeling of isogeometric analysis and propose a new method for its coupling with meshfree discretizations. We exploit the strengths of both methods by using IGA to provide a smooth, geometrically-exact surface discretization of the problem domain boundary, while the Reproducing Kernel Particle Method (RKPM) discretization is used to provide the volumetric discretization of the domain interior. The coupling strategy is based upon the higher-order consistency or reproducing conditions that are directly imposed in the physical domain. The resulting coupled method enjoys several favorable features: (i) it preserves the geometric exactness of IGA, (ii) it circumvents the need for global volumetric parameterization of the problem domain, (iii) it achieves arbitrary-order approximation accuracy while preserving higher-order smoothness of the discretization. Several numerical examples are solved to show the optimal convergence properties of the coupled IGA–RKPM formulation, and to demonstrate its effectiveness in constructing volumetric discretizations for complex-geometry objects.
As for the next contribution, we exploit the use of smooth, high-order spline basis functions in IGA to solve high-order surface PDEs governing the morphological evolution of vesicles. These governing equations are often consisted of geometric PDEs, high-order PDEs on stationary or evolving surfaces, or a combination of them. We propose an isogeometric formulation for solving these PDEs. In the context of geometric PDEs, we consider phase-field approximations of mean curvature flow and Willmore flow problems and numerically study the convergence behavior of isogeometric analysis for these problems. As a model problem for high-order PDEs on stationary surfaces, we consider the Cahn–Hilliard equation on a sphere, where the surface is modeled using a phase-field approach. As for the high-order PDEs on evolving surfaces, a phase-field model of a deforming multi-component vesicle, which consists of two fourth-order nonlinear PDEs, is solved using the isogeometric analysis in a primal variational framework. Through several numerical examples in 2D, 3D and axisymmetric 3D settings, we show the robustness of IGA for solving the considered phase-field models.
Finally, we present a monolithic, implicit formulation based on isogeometric analysis and generalized-alpha time integration for simulating hydrodynamics of vesicles according to a phase-field model. Compared to earlier works, the number of equations of the phase-field model which need to be solved is reduced by leveraging high continuity of NURBS functions, and the algorithm is extended to 3D settings. We use residual-based variational multi-scale method (RBVMS) for solving Navier–Stokes equations, while the rest of PDEs in the phase-field model are treated using a standard Galerkin-based IGA. We introduce the resistive immersed surface (RIS) method into the formulation which can be employed for an implicit description of complex geometries using a diffuse-interface approach. The implementation highlights the robustness of the RBVMS method for Navier–Stokes equations of incompressible flows with non-trivial localized forcing terms including bending and tension forces of the vesicle. The potential of the phase-field model and isogeometric analysis for accurate simulation of a variety of fluid-vesicle interaction problems in 2D and 3D is demonstrated.