TY - JOUR A1 - Engelmann, Ulrich M. A1 - Simsek, Beril A1 - Shalaby, Ahmed A1 - Krause, Hans-Joachim T1 - Key contributors to signal generation in frequency mixing magnetic detection (FMMD): an in silico study JF - Sensors N2 - Frequency mixing magnetic detection (FMMD) is a sensitive and selective technique to detect magnetic nanoparticles (MNPs) serving as probes for binding biological targets. Its principle relies on the nonlinear magnetic relaxation dynamics of a particle ensemble interacting with a dual frequency external magnetic field. In order to increase its sensitivity, lower its limit of detection and overall improve its applicability in biosensing, matching combinations of external field parameters and internal particle properties are being sought to advance FMMD. In this study, we systematically probe the aforementioned interaction with coupled Néel–Brownian dynamic relaxation simulations to examine how key MNP properties as well as applied field parameters affect the frequency mixing signal generation. It is found that the core size of MNPs dominates their nonlinear magnetic response, with the strongest contributions from the largest particles. The drive field amplitude dominates the shape of the field-dependent response, whereas effective anisotropy and hydrodynamic size of the particles only weakly influence the signal generation in FMMD. For tailoring the MNP properties and parameters of the setup towards optimal FMMD signal generation, our findings suggest choosing large particles of core sizes dc > 25 nm nm with narrow size distributions (σ < 0.1) to minimize the required drive field amplitude. This allows potential improvements of FMMD as a stand-alone application, as well as advances in magnetic particle imaging, hyperthermia and magnetic immunoassays. KW - key performance indicators KW - magnetic biosensing KW - coupled Néel–Brownian relaxation dynamics KW - frequency mixing magnetic detection KW - magnetic relaxation KW - micromagnetic simulation KW - magnetic nanoparticles Y1 - 2024 U6 - https://doi.org/10.3390/s24061945 SN - 1424-8220 N1 - This article belongs to the Special Issue "Advances in Magnetic Sensors and Their Applications" VL - 24 IS - 6 PB - MDPI CY - Basel ER - TY - JOUR A1 - Zhen, Manghao A1 - Liang, Yunpei A1 - Staat, Manfred A1 - Li, Quanqui A1 - Li, Jianbo T1 - Discontinuous fracture behaviors and constitutive model of sandstone specimens containing non-parallel prefabricated fissures under uniaxial compression JF - Theoretical and Applied Fracture Mechanics N2 - The deformation and damage laws of non-homogeneous irregular structural planes in rocks are the basis for studying the stability of rock engineering. To investigate the damage characteristics of rock containing non-parallel fissures, uniaxial compression tests and numerical simulations were conducted on sandstone specimens containing three non-parallel fissures inclined at 0°, 45° and 90° in this study. The characteristics of crack initiation and crack evolution of fissures with different inclinations were analyzed. A constitutive model for the discontinuous fractures of fissured sandstone was proposed. The results show that the fracture behaviors of fissured sandstone specimens are discontinuous. The stress–strain curves are non-smooth and can be divided into nonlinear crack closure stage, linear elastic stage, plastic stage and brittle failure stage, of which the plastic stage contains discontinuous stress drops. During the uniaxial compression test, the middle or ends of 0° fissures were the first to crack compared to 45° and 90° fissures. The end with small distance between 0° and 45° fissures cracked first, and the end with large distance cracked later. After the final failure, 0° fissures in all specimens were fractured, while 45° and 90° fissures were not necessarily fractured. Numerical simulation results show that the concentration of compressive stress at the tips of 0°, 45° and 90° fissures, as well as the concentration of tensile stress on both sides, decreased with the increase of the inclination angle. A constitutive model for the discontinuous fractures of fissured sandstone specimens was derived by combining the logistic model and damage mechanic theory. This model can well describe the discontinuous drops of stress and agrees well with the whole processes of the stress–strain curves of the fissured sandstone specimens. KW - Constitutive model KW - Damage mechanics theory KW - Discontinuous fractures KW - Uniaxial compression test KW - Non-parallel fissures Y1 - 2024 U6 - https://doi.org/10.1016/j.tafmec.2024.104373 SN - 0167-8442 VL - 131 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Pieronek, Lukas A1 - Kleefeld, Andreas T1 - On trajectories of complex-valued interior transmission eigenvalues JF - Inverse problems and imaging : IPI N2 - This paper investigates the interior transmission problem for homogeneous media via eigenvalue trajectories parameterized by the magnitude of the refractive index. In the case that the scatterer is the unit disk, we prove that there is a one-to-one correspondence between complex-valued interior transmission eigenvalue trajectories and Dirichlet eigenvalues of the Laplacian which turn out to be exactly the trajectorial limit points as the refractive index tends to infinity. For general simply-connected scatterers in two or three dimensions, a corresponding relation is still open, but further theoretical results and numerical studies indicate a similar connection. KW - Interior transmission problem KW - Eigenvalue trajectories KW - Complex-valued eigenvalues Y1 - 2024 U6 - https://doi.org/10.3934/ipi.2023041 SN - 1930-8337 (Print) SN - 1930-8345 (Online) VL - 18 IS - 2 SP - 480 EP - 516 PB - AIMS CY - Springfield, Mo ER - TY - JOUR A1 - Ayala, Rafael Ceja A1 - Harris, Isaac A1 - Kleefeld, Andreas T1 - Direct sampling method via Landweber iteration for an absorbing scatterer with a conductive boundary JF - Inverse Problems and Imaging N2 - In this paper, we consider the inverse shape problem of recovering isotropic scatterers with a conductive boundary condition. Here, we assume that the measured far-field data is known at a fixed wave number. Motivated by recent work, we study a new direct sampling indicator based on the Landweber iteration and the factorization method. Therefore, we prove the connection between these reconstruction methods. The method studied here falls under the category of qualitative reconstruction methods where an imaging function is used to recover the absorbing scatterer. We prove stability of our new imaging function as well as derive a discrepancy principle for recovering the regularization parameter. The theoretical results are verified with numerical examples to show how the reconstruction performs by the new Landweber direct sampling method. Y1 - 2024 U6 - https://doi.org/10.3934/ipi.2023051 SN - 1930-8337 SN - 1930-8345 (eISSN) VL - 18 IS - 3 SP - 708 EP - 729 PB - AIMS CY - Springfield ER - TY - JOUR A1 - Harris, Isaac A1 - Kleefeld, Andreas T1 - The inverse scattering problem for a conductive boundary condition and transmission eigenvalues JF - Applicable Analysis N2 - In this paper, we consider the inverse scattering problem associated with an inhomogeneous media with a conductive boundary. In particular, we are interested in two problems that arise from this inverse problem: the inverse conductivity problem and the corresponding interior transmission eigenvalue problem. The inverse conductivity problem is to recover the conductive boundary parameter from the measured scattering data. We prove that the measured scatted data uniquely determine the conductivity parameter as well as describe a direct algorithm to recover the conductivity. The interior transmission eigenvalue problem is an eigenvalue problem associated with the inverse scattering of such materials. We investigate the convergence of the eigenvalues as the conductivity parameter tends to zero as well as prove existence and discreteness for the case of an absorbing media. Lastly, several numerical and analytical results support the theory and we show that the inside–outside duality method can be used to reconstruct the interior conductive eigenvalues. KW - Transmission eigenvalues KW - Conductive boundary condition KW - Inverse scattering Y1 - 2018 U6 - https://doi.org/10.1080/00036811.2018.1504028 SN - 1563-504X VL - 99 IS - 3 SP - 508 EP - 529 PB - Taylor & Francis CY - London ER - TY - JOUR A1 - Kleefeld, Andreas A1 - Pieronek, J. T1 - Elastic transmission eigenvalues and their computation via the method of fundamental solutions JF - Applicable Analysis N2 - A stabilized version of the fundamental solution method to catch ill-conditioning effects is investigated with focus on the computation of complex-valued elastic interior transmission eigenvalues in two dimensions for homogeneous and isotropic media. Its algorithm can be implemented very shortly and adopts to many similar partial differential equation-based eigenproblems as long as the underlying fundamental solution function can be easily generated. We develop a corroborative approximation analysis which also implicates new basic results for transmission eigenfunctions and present some numerical examples which together prove successful feasibility of our eigenvalue recovery approach. KW - elastic scattering KW - method of fundamental solutions KW - Interior transmission eigenvalues Y1 - 2020 U6 - https://doi.org/10.1080/00036811.2020.1721473 SN - 1563-504X VL - 100 IS - 16 SP - 3445 EP - 3462 PB - Taylore & Francis CY - London ER - TY - JOUR A1 - Breuß, Michael A1 - Kleefeld, Andreas T1 - Implicit monotone difference methods for scalar conservation laws with source terms JF - Acta Mathematica Vietnamica N2 - In this article, a concept of implicit methods for scalar conservation laws in one or more spatial dimensions allowing also for source terms of various types is presented. This material is a significant extension of previous work of the first author (Breuß SIAM J. Numer. Anal. 43(3), 970–986 2005). Implicit notions are developed that are centered around a monotonicity criterion. We demonstrate a connection between a numerical scheme and a discrete entropy inequality, which is based on a classical approach by Crandall and Majda. Additionally, three implicit methods are investigated using the developed notions. Next, we conduct a convergence proof which is not based on a classical compactness argument. Finally, the theoretical results are confirmed by various numerical tests. KW - Entropy solution KW - Source term KW - Monotone methods KW - Implicit methods KW - Finite difference methods KW - Conservation laws Y1 - 2020 U6 - https://doi.org/10.1007/s40306-019-00354-1 SN - 2315-4144 N1 - Corresponding author: Andreas Kleefeld VL - 45 SP - 709 EP - 738 PB - Springer Singapore CY - Singapore ER - TY - JOUR A1 - Asante-Asamani, E.O. A1 - Kleefeld, Andreas A1 - Wade, B.A. T1 - A second-order exponential time differencing scheme for non-linear reaction-diffusion systems with dimensional splitting JF - Journal of Computational Physics N2 - A second-order L-stable exponential time-differencing (ETD) method is developed by combining an ETD scheme with approximating the matrix exponentials by rational functions having real distinct poles (RDP), together with a dimensional splitting integrating factor technique. A variety of non-linear reaction-diffusion equations in two and three dimensions with either Dirichlet, Neumann, or periodic boundary conditions are solved with this scheme and shown to outperform a variety of other second-order implicit-explicit schemes. An additional performance boost is gained through further use of basic parallelization techniques. KW - Exponential time differencing KW - Real distinct pole KW - Dimensional splitting KW - Reaction-diffusion systems KW - Matrix exponential Y1 - 2020 U6 - https://doi.org/10.1016/j.jcp.2020.109490 SN - 0021-9991 N1 - Corresponding author: Andreas Kleefeld VL - 415 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Martín-Vaquero, J. A1 - Kleefeld, Andreas T1 - Solving nonlinear parabolic PDEs in several dimensions: Parallelized ESERK codes JF - Journal of Computational Physics N2 - There is a very large number of very important situations which can be modeled with nonlinear parabolic partial differential equations (PDEs) in several dimensions. In general, these PDEs can be solved by discretizing in the spatial variables and transforming them into huge systems of ordinary differential equations (ODEs), which are very stiff. Therefore, standard explicit methods require a large number of iterations to solve stiff problems. But implicit schemes are computationally very expensive when solving huge systems of nonlinear ODEs. Several families of Extrapolated Stabilized Explicit Runge-Kutta schemes (ESERK) with different order of accuracy (3 to 6) are derived and analyzed in this work. They are explicit methods, with stability regions extended, along the negative real semi-axis, quadratically with respect to the number of stages s, hence they can be considered to solve stiff problems much faster than traditional explicit schemes. Additionally, they allow the adaptation of the step length easily with a very small cost. Two new families of ESERK schemes (ESERK3 and ESERK6) are derived, and analyzed, in this work. Each family has more than 50 new schemes, with up to 84.000 stages in the case of ESERK6. For the first time, we also parallelized all these new variable step length and variable number of stages algorithms (ESERK3, ESERK4, ESERK5, and ESERK6). These parallelized strategies allow to decrease times significantly, as it is discussed and also shown numerically in two problems. Thus, the new codes provide very good results compared to other well-known ODE solvers. Finally, a new strategy is proposed to increase the efficiency of these schemes, and it is discussed the idea of combining ESERK families in one code, because typically, stiff problems have different zones and according to them and the requested tolerance the optimum order of convergence is different. KW - Multi-dimensional partial differential equations KW - Higher-order codes KW - Nonlinear PDEs Y1 - 2020 U6 - https://doi.org/10.1016/j.jcp.2020.109771 SN - 0021-9991 IS - 423 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Kleefeld, Andreas T1 - The hot spots conjecture can be false: some numerical examples JF - Advances in Computational Mathematics N2 - The hot spots conjecture is only known to be true for special geometries. This paper shows numerically that the hot spots conjecture can fail to be true for easy to construct bounded domains with one hole. The underlying eigenvalue problem for the Laplace equation with Neumann boundary condition is solved with boundary integral equations yielding a non-linear eigenvalue problem. Its discretization via the boundary element collocation method in combination with the algorithm by Beyn yields highly accurate results both for the first non-zero eigenvalue and its corresponding eigenfunction which is due to superconvergence. Additionally, it can be shown numerically that the ratio between the maximal/minimal value inside the domain and its maximal/minimal value on the boundary can be larger than 1 + 10− 3. Finally, numerical examples for easy to construct domains with up to five holes are provided which fail the hot spots conjecture as well. KW - Numerics KW - Boundary integral equations KW - Potential theory KW - Helmholtz equation KW - Interior Neumann eigenvalues Y1 - 2021 U6 - https://doi.org/10.1007/s10444-021-09911-5 SN - 1019-7168 VL - 47 PB - Springer CY - Dordrecht ER -