Lukas Pflug

• Dr
• Senior Researcher at Friedrich-Alexander-University Erlangen-Nürnberg

Structural coloration abounds in nature and its remarkable optical effects are mimicked in synthetic photonic crystals and glasses. However, the color saturation of these synthetic structures is often diminished by incoherent scattering caused by defects and irregularities. The inclusion of absorbing materials increases color saturation, but where...

We consider a class of nonlocal conservation laws with exponential kernel and prove that quantities involving the nonlocal term [Formula: see text] satisfy an Oleĭnik-type entropy condition. More precisely, under different sets of assumptions on the velocity function [Formula: see text], we prove that [Formula: see text] satisfies a one-sided Lipsc...

The transition matrix, frequently abbreviated as T-matrix, contains the complete information in a linear approximation of how a spatially localized object scatters an incident field. The T-matrix is used to study the scattering response of an isolated object and describes the optical response of complex photonic materials made from ensembles of ind...

Topology optimization under uncertainty or reliability-based topology optimization is usually numerically very expensive. This is mainly due to the fact that an accurate evaluation of the probabilistic model requires the system to be simulated for a large number of varying parameters. Traditional gradient-based optimization schemes thus face the di...

https://zenodo.org/records/11264121

Plasmonic nanoparticles have intriguing optical properties which make them suitable candidates for sensing or theranostic applications. Anisotropic patchy particles, where metal is locally deposited on the surface of a core particle, exhibit plasmon resonances that can be specifically adjusted for these applications. However, many existing synthesi...

The transition matrix, frequently abbreviated as T-matrix, contains the complete information in a linear approximation of how a spatially localized object scatters an incident field. The T-matrix is used to study the scattering response of an isolated object and describes the optical response of complex photonic materials made from ensembles of ind...

This paper focuses on the topology optimization of a broadband acoustic transition section that connects two cylindrical waveguides with different radii. The primary objective is to design a transition section that maximizes the transmission of a planar acoustic wave while ensuring that the transmitted wave exhibits a planar shape. Helmholtz equati...

Nanostructured materials that mimic structural coloration in nature can be synthetically created by colloidal self‐assembly. To maximize optical effects, the natural world integrates melanin as a broadband absorber to remove incoherently scattered light. Polydopamine (PDA) is used as a synthetic analog of natural melanin to systematically investiga...

The large-scale synthesis of nanoparticles (NPs) with defined properties requires detailed understanding of the underlying formation mechanisms and kinetics. The formation mechanisms of bimetallic NPs are still not sufficiently understood due to the complex reaction chemistry, which makes the control of the supersaturation as the thermodynamic driv...

This research article focuses on the targeted color design of silver–gold alloy nanoparticles (NPs), employing a multivariate optimization approach. NP synthesis involves interconnected process parameters, making independent variation challenging. Data-based property–process relationships are established to optimize optical properties effectively....

In this contribution, we study the singular limit problem of a nonlocal conservation law with a discontinuity in space. The corresponding local equation can be transformed diffeomorphically to a classical scalar conservation law to which the well-known Kružkov theory can be applied. However, the nonlocal equation does not scale that way, which is w...

In this contribution, we present a numerical analysis of the continuous stochastic gradient (CSG) method, including applications from topology optimization and convergence rates. In contrast to standard stochastic gradient optimization schemes, CSG does not discard old gradient samples from previous iterations. Instead, design dependent integration...

In this contribution, we present a full overview of the continuous stochastic gradient (CSG) method, including convergence results, step size rules and algorithmic insights. We consider optimization problems in which the objective function requires some form of integration, e.g., expected values. Since approximating the integration by a fixed quadr...

The digital transformation and consequent use of new digital technologies not only have a substantial impact on society and companies, but also on science. Analog documentation as we have known it for centuries will eventually be replaced by intelligent and FAIR (Findable, Accessible, Interoperable, and Reusable) systems. In addition to the actual...

We prove the convergence of solutions of nonlocal conservation laws to their local entropic counterpart for a fundamentally extended class of nonlocal kernels when these kernels approach a Dirac distribution. The nonlocal kernels are assumed to have fixed support and do not have to be monotonic. With sharp estimates of the nonlocal kernels and a su...

We study the long-time behaviour of the unique weak solution of a nonlocal regularisation of the (inviscid) Burgers equation where the velocity is approximated by a one-sided convolution with an exponential kernel. The initial datum is assumed to be positive, bounded, and integrable. The asymptotic profile is given by the ‘ N -wave’ entropy solutio...

Despite great progress in the synthetic chemistry of InP QDs, a predictive model to describe their temporal formation is still missing. In this work, we introduce a population balance model incorporating liquid phase reactions, homogeneous nucleation and reaction-limited growth of InP supported with the highly reproducible and reliable experimental...

In this work, we propose a robust optimization approach to mitigate the impact of uncertainties in particle precipitation. Our model incorporates partial differential equations, more particular nonlinear and nonlocal population balance equations to describe particle synthesis. The goal of the optimization problem is to design products with desired...

The unique properties of anisotropic and composite particles are increasingly being leveraged in modern particulate products. However, tailored synthesis of particles characterized by multi-dimensional dispersed properties remains in its infancy and few mathematical models for their synthesis exist. Here, we present a novel, accurate and highly eff...

We consider a class of nonlocal conservation laws with exponential kernel and prove that quantities involving the nonlocal term $W:=\mathbb{1}_{(-\infty,0]}(\cdot)\exp(\cdot) \ast \rho$ satisfy an Ole\u{\i}nik-type entropy condition. More precisely, under different sets of assumptions on the velocity function $V$, we prove that $W$ satisfies a one-...

We propose a novel approach to optimize the design of heterogeneous materials, with the goal of enhancing their effective fracture toughness under mode-I loading. The method employs a Gaussian processes-based Bayesian optimization framework to determine the optimal shapes and locations of stiff elliptical inclusions within a periodic microstructure...

Optical properties of nanoparticles largely depend on their shape and material distribution. Given any such property, e.g., a desired color, the aim is to design an optimal nanoparticle, whose properties best match these targets. The corresponding nonlinear optimization problem is challenging due to numerous difficulties: The objective function is...

In this contribution, we present a numerical analysis of the continuous stochastic gradient (CSG) method, including applications from topology optimization and convergence rates. In contrast to standard stochastic gradient optimization schemes, CSG does not discard old gradient samples from previous iterations. Instead, design dependent integration...

Multidimensional particle properties determine the product properties in numerous advanced applications. Accurate and statistically meaningful measurements of complex particles and their multidimensional distributions are highly challenging but strongly needed. 2D particle size distributions of plasmonic nanoparticles of complex regular shape can b...

In this work we present a nonlocal conservation law with a velocity depending on an integral term over a part of the space. The model class covers already existing models in literature, but it is also able to describe new dynamics mainly arising in the context of traffic flow modelling. We prove the existence and uniqueness of weak solutions of the...

The optimal design of nanoparticles with respect to their optical properties is one of the main foci within nanoparticle technology. In this contribution, we suggest a new design optimization method in the framework of which the discrete dipole approximation (DDA) is used to approximate the solution of Maxwell’s equation in time-harmonic form. In t...

We deal with the problem of approximating a scalar conservation law by a conservation law with nonlocal flux. As convolution kernel in the nonlocal flux, we consider an exponential-type approximation of the Dirac distribution. We then obtain a total variation bound on the nonlocal term and can prove that the (unique) weak solution of the nonlocal p...

Recent developments in the field of computational modeling of fracture have opened up possibilities for designing structures against failure. A special case, called interfacial fracture or delamination, can occur in loaded composite structures where two or more materials are bonded together at comparatively weak interfaces. Due to the potential cra...

In this note, we extend the known results on the existence and uniqueness of weak solutions to conservation laws with nonlocal flux. In case the nonlocal term is given by a convolution γ∗q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usep...

We consider conservation laws with nonlocal velocity and show for nonlocal weights of exponential type that the unique solutions converge in a weak or strong sense (dependent on the regularity of the velocity) to the entropy solution of the local conservation law when the nonlocal weight approaches a Dirac distribution. To this end, we establish fi...

Recent developments in the field of computational modeling of fracture have opened up possibilities for designing structures against failure. A special case, called interfacial fracture or delamination, can occur in loaded composite structures where two or more materials are bonded together at comparatively weak interfaces. Due to the potential cra...

In this study, we introduce a method for the simultaneous retrieval of two-dimensional size-composition distributions of noble metal Ag-Au alloy nanoparticles utilizing an analytical ultracentrifuge equipped with a multiwavelength extinction detector (MWL-AUC). MWL-AUC is used to measure coupled optical and sedimentation properties of the particles...

The recently proposed continuous stochastic gradient descent (CSG) scheme represents an efficient method for the solution of a large class of stochastic optimization problems. During the course of the iterations, CSG uses a tailored convex combination of previously drawn gradient and objective function samples to approximate the full gradient and o...

Knowledge-based determination of the best-possible experimental setups for nanoparticle design is highly challenging. Additionally, such processes are accompanied by noticeable uncertainties. Therefore, protection against those is needed. Robust optimization helps determining optimal processes. The latter guarantees quality requirements regardless...

A recent article introduced thecontinuous stochastic gradient method (CSG) for the efficient solution of a class of stochastic optimization problems. While the applicability of known stochastic gradient type methods is typically limited to expected risk functions, no such limitation exists for CSG. This advantage stems from the computation of desig...

We study nonlocal conservation laws with a discontinuous flux function of regularity L^{\infty} (R) in the spatial variable and show existence and uniqueness of weak solutions in C ([0, T ] ; L^{1}_{loc}(\R)) , as well as related maximum principles. We achieve this well-posedness by a proper reformulation in terms of a fixed-point problem. This fix...

We consider a class of nonlocal conservation laws with a second-order viscous regularization term which finds an application in modelling macroscopic traffic flow. The velocity function depends on a weighted average of the density ahead, where the averaging kernel is of exponential type. We show that, as the nonlocal impact and the viscosity parame...

Anisotropic nanoparticles offer considerable promise for applications but also present significant challenges in terms of their characterization. Recent developments in the electroless deposition of silver patches directly onto colloidal silica particles have opened up a simple and scalable synthesis method for patchy particles with tunable optical...

Advances in the computational modeling of fracture of solids have opened up new possibilities for structural design optimization to enhance fracture properties. Here, we investigate material optimization and design to improve fracture behavior of composite structures under quasi-static conditions. The rate-independent structural problem considering...

This article provides mathematical proof of the existence of stationary solutions for the coagulation equation including source and efflux terms. We demonstrate the convergence of time dependent solutions to these stationary solutions and highlight the exponential rate of convergence. These properties are analyzed for affine linear coagulation kern...

We study the exact boundary controllability of a class of nonlocal conservation laws modeling traffic flow. The velocity of the macroscopic dynamics depends on a weighted average of the traffic density ahead and the averaging kernel is of exponential type. Under specific assumptions, we show that the boundary controls can be used to steer the syste...

Knowledge-based determination of the best-possible experimental setups for nanoparticle design is highly challenging.Additionally, such processes are accompanied by noticeable uncertainties. Therefore, protection against those is needed. Robust optimization helps determining optimal processes. The latter guarantees quality requirements regardless o...

In this note, we extend the known results on the existence and uniqueness of weak solutions to conservation laws with nonlocal flux. In case the nonlocal term is given by a convolution γ * q, we weaken the standard assumption on the kernel γ ∈ L ∞ (0, T); W 1,∞ (R) to the substantially more general condition γ ∈ L ∞ ((0, T); BV (R)), which allows f...

We deal with the problem of approximating a scalar conservation law by a conservation law with nonlocal flux. As convolution kernel in the nonlocal flux, we consider an exponential-type approximation of the Dirac distribution. This enables us to obtain a total variation bound on the nonlocal term. By using this, we prove that the (unique) weak solu...

We consider a class of nonlocal conservation laws with a second-order viscous regularization term which finds an application in modelling macroscopic traffic flow. The velocity function depends on a weighted average of the density ahead, where the averaging kernel is of exponential type. We show that, as the nonlocal reach and the viscosity paramet...

In this contribution, we study the existence and uniqueness of nonlocal transport equations. The term "nonlocal" refers to the fact that the flux function's derivative will be integrated over a neighborhood of the corresponding space-time coordinate. We will demonstrate existence and uniqueness of weak solutions for T V \cap L \infty initial datum...

We consider a system of nonlocal balance laws where every single balance law is coupled with the remaining ones by a nonlocal velocity function which takes into account the averaged density of all other equations as well as by a right hand "semi-linear" term. We show existence and uniqueness of weak solutions for small time horizon and a maximum pr...

We study the exact boundary controllability of a class of nonlocal conservation laws modeling traffic flow. The velocity of the macroscopic dynamics depends on a weighted average of the traffic density ahead and the averaging kernel is of exponential type. Under specific assumptions, we show that the boundary controls can be used to steer the syste...

The COVID-19 pandemic has led to an unprecedented world-wide effort to gather data, model, and understand the viral spread. Entire societies and economies are desperate to recover and get back to normality. However, to this end accurate models are of essence that capture both the viral spread and the courses of disease in space and time at reasonab...

Processes in the field of chemical engineering do not consist of one single step, but typically a high number of strongly interconnected unit operations linked with recycling streams. This inherent complexity further exacerbates when distributed particle properties, i.e., dispersity, must be considered, noteworthy being the case whenever particulat...

This paper presents a novel method for the solution of a particular class of structural optimzation problems: the continuous stochastic gradient method (CSG). In the simplest case, we assume that the objective function is given as an integral of a desired property over a continuous parameter set. The application of a quadrature rule for the approxi...

The spread of COVID-19 at the end of 2019 with its delay properties requires improved dynamical models to forecast and evaluate specific measures in politics. As these policy measures require real-time information on the evolution of the diseases, quantities like delay due to incubation time and infectious period can not be neglected and need to be...

In order to obtain high‐quality particulate products with tailored properties, process conditions and their evolution in time must be chosen appropriately. Although the efficiency of these products depends on their dispersity in several dimensions, in established processes particle size is usually the decisive variable to adjust. As part of the syn...

In this study we present a reformulation for a broad class of population balance equations that model nucleation and size dependent growth. This formulation enables the definition of new numerical methods, which have two advantages compared to existing schemes in the literature (e.g. finite volume type methods and methods based on the evolution of...

This study considers nonlocal conservation laws in which the velocity depends nonlocally on the solution not in real time but in a time-delayed manner. Nonlocal refers to the fact that the velocity of the conservation law depends on the solution integrated over a specific area in space. In every model modelling human’s behavior a time delay as reac...

In this contribution we present a reformulation of population balance equations which model nucleation and size dependent growth. This formulation allows to define new numerical methods, significantly superior to already existent schemes in the literature in the following way: i) Higher precision due to non-smoothing ii) less run-time in comparison...

Particle science and technology evolve toward ever increasing complexity with respect to the multidimensional particle properties of size, shape, surface, internal structure, and composition. In this study, the theoretical background is elaborated for multidimensional particle size distributions (PSDs) by transferring the concepts known from 1D siz...

The simultaneous precipitation of multiple solid phases inside a T-mixer was investigated both numerically and experimentally based on the model system BaSO4 and BaCO3. With a population balance equation model including mixing, hydrochemistry and solid formation kinetics, the global solid phase composition was calculated and compared to experimenta...

We show that for monotone initial datum the solution of nonlocal conservation laws converges to the entropy solution of the corresponding local conservation laws when the nonlocal reach tends to zero. This particularly covers the principle cases of conservation laws: shocks and rarefactions. The considered problem is addressed by studying the Entro...

We consider a nonlocal conservation law on a bounded spatial domain and show existence and uniqueness of weak solutions for nonnegative flux function and left-hand-side boundary datum. The nonlocal term is located in the flux function of the conservation law, averaging the solution by means of an integral at every spatial coordinate and every time,...

For most particle-based applications, formulation in the liquid phase is a decisive step, and thus, particle interactions and stability in liquid media are of major importance. The concept of Hansen solubility parameters (HSP) was initially invented to describe the interactions of (polymer) molecules and their solubility in different liquids and is...

Properties of nanoparticles are influenced by various parameters like size, shape or composition. Comprehensive high throughput characterization techniques are urgently needed to improve synthesis, scale up to production and make way for new applications of multidimensional particulate systems. In this study, we present a method for measuring two-d...

In this article, we generalize some existence and uniqueness results in [22] for scalar nonlocal balance laws to multi-dimensional nonlocal balance laws. Also in the multi-dimensional case it is possible to get rid of the Entropy condition which is usually postulated to guarantee a unique weak solution of (local) balance laws but which we prove to...

Nichtlokale Bilanzgleichungen sind nichtlineare partielle Integro-Differentialgleichungen, die im Rahmen der Modellierung realer Abläufe vielseitig anwendbar sind. Von der Beschreibung von Wachstumsprozessen in der Nanopartikelsynthese bis hin zu makroskopischen Modellierungen von Verkehrsfluss sind Gleichungen diesen Typs von entscheidender Bedeut...

A class of algorithms for the solution of discrete material optimization problems in electromagnetic applications is discussed. The idea behind the algorithm is similar to that of the sequential programming. However, in each major iteration a model is established on the basis of an appropriately parametrized material tensor. The resulting nonlinear...

A class of algorithms for the solution of discrete material optimization problems in electromagnetic applications is discussed. The idea behind the algorithm is similar to that of the sequential programming. However, in each major iteration a model is established on the basis of an appropriately parametrized material tensor. The resulting nonlinear...

We consider a class of nonlocal balance laws as initial value problems on a finite time horizon and show existence and uniqueness of the corresponding weak solutions. The description “nonlocal” refers to the velocity of the balance law that depends on the weighted integral over an area in space at any given time. Existence of a weak solution for in...

Mesocrystalline particles have been recognized as a class of multifunctional materials with potential applications in different fields. However, the internal organization of nanocomposite mesocrystals and its influence on the final properties have not yet been investigated. In this paper, a novel strategy based on electrodynamic simulations is deve...

Dielectric-metal core-shell particles with morphologically tunable optical properties are highly promising candidates for applications ranging from theranostics, energy harvesting and storage to pigments and sensors. Most structures of interest have, until now, been produced in small volume wet chemical batch approaches which are difficult to scale...

Unit operations and product design are the two most important pillars of chemical engineering. Product design is the formation, formulation, handling, manufacturing, and characterization of complex multiphase products with specific properties and is thus at the core of mesoscale science and engineering. The applications define the required product...

We consider shape optimization for objects illuminated by light. More precisely, we focus on time-harmonic solutions of the Maxwell system in curl-curl-form scattered by an arbitrary shaped rigid object. Given a class of cost functionals, including the scattered energy and the extinction cross section, we develop an adjoint-based shape optimization...

This work presents the application of a Fully Implicit Method for Ostwald Ripening (FIMOR) for simulating the ripening of ZnO quantum dots (QDs). Its stable numerics allow FIMOR to employ the full exponential term of the Gibbs–Thomson equation which significantly outperforms the common Taylor-approximation at typical QD sizes below 10 nm. The imple...

Small particles are exploited for their optical properties in a vast array of applications such as paint, inks, textiles and cosmetics. In many of these application areas, features of the particles (size, shape and topology) can be on the nanoscale, a fact which is often fundamentally important to their desirable properties. With the broad range of...

In this paper, a new Ansatz for modelling the Baculovirus infection cycle is presented. The base of this model is the cell cycle distribution at the time of infection. It is possible to calculate the growth of the culture and the initiation of virus processing by considering cell cycle distribution. By taking into account the length of the viral ge...