A two-field computational model couples cellular brain development with cortical folding - Zarzor, Mohammad Saeed; Käßmair, Stefan; Steinmann, Paul; Blümcke, Ingmar; Budday, Silvia (Brain Multiphysics, 2021)
The convoluted macroscopic shape of the mammalian brain plays an important role for brain function. To date, the link between the cellular processes during brain development and normal or abnormal cortical folding on the macroscopic scale remains insufficiently understood. Disruption of cellular division, migration, or connectivity may lead to malformations of cortical development associated with neurological disorders like schizophrenia, autism, or epilepsy. Here, we use a computational model, which couples an advection-diffusion model with finite growth, to assess the link between cellular division and migration on the cell scale and growth and cortical folding on the tissue or organ scale. It introduces the cell density as independent field controlling volumetric growth. This allows us to numerically study the influence of cell migration velocity, cell diffusivity, and the temporally changing local stiffness of brain tissue on the cortical folding process during normal brain development. We show that the model is capable of capturing the local distribution of cells through the comparison with histologically stained sections of the developing human brain. Our results further demonstrate that it is important to take temporal changes in tissue stiffness into account, which naturally occur during brain development. The present study constitutes an important step towards a computational model that could help to better understand, diagnose, and, eventually, treat neurological disorders arising from abnormal cellular development and cortical malformations.<br
Keywords: Brain development; Cortical folding; Neurogenesis; Finite growth; Multifield modeling; Mechanical instability

Comparative computational analysis of the Cahn–Hilliard equation with emphasis on C1-continuous methods - Käßmair, Stefan; Steinmann, Paul (Journal of Computational Physics, 2016)
The numerical treatment of the fourth-order Cahn–Hilliard equation is nonstandard. Using a Galerkin -method necessitates, for instance, piecewise smooth and globally C1-continuous basis functions or a mixed formulation. The latter is obtained introducing an auxiliary field which allows to rephrase the Cahn–Hilliard equation as a set of two coupled second-order equations. In view of this, the formulation in terms of the primal unknown appears to be a more intuitive and natural choice but requires a C1-continuous interpolation. Therefore, isogeometric analysis, using a spline basis, and natural element analysis are addressed in the present contribution. Mixed second-order finite element methods introducing the chemical potential or alternatively a nonlocal concentration as auxiliary field serve as references to which both higher-order methods are compared in terms of accuracy and efficiency.
EAM Import::2019-03-13

Computational first-order homogenization in chemo-mechanics - Käßmair, Stefan; Steinmann, Paul (Archive of Applied Mechanics, 2018)
In the present contribution, we consider species diffusion coupled to finite deformations in strongly heterogeneous microstructures. A semi-dual energy formulation parameterized in terms of the chemical potential is obtained by Legendre transformation of the free energy. Doing so avoids the presence of higher gradients of the deformation field. The constitutive response at the macroscopic level is obtained using variationally consistent homogenization (Larson et al. in Int J Numer Method Eng 81(13):1659–1686, 2010. doi:10.1002/nme.2747). This approach allows to treat transient microscale problems on a representative volume element which has a finite size, i.e., the scales are not clearly separated. Full details of the implementation are provided. A series of numerical examples compares the homogenization results to single-scale formulations which fully resolve all microstructural features.
CRIS-Team Scopus Importer:2022-08-05
Keywords: Coupled problem; First-order homogenization; Transient microscale

Exploring the interplay between cellular development and mechanics in the developing human brain - Zarzor, Mohammad Saeed; Käßmair, Stefan; Steinmann, Paul; Blümcke, Ingmar; Budday, Silvia (Proceedings in Applied Mathematics and Mechanics, 2021)
The human brain has a complex structure on both cellular and organ scales. This structure is closely related to the brain's abilities and functions. Disruption of one of the biological processes occurring during brain development on the cellular scale may affect the cortical folding pattern of the brain on the organ scale. However, the link between disruptions in cellular brain development and associated cortical malformation remains largely unknown. From a mechanical perspective, the forces generated during development lead to mechanical instability and, eventually, the mergence of cortical folds. To fully understand mechanism underlying malformations of cortical development, it is key to consider both the events that occur on the cellular scale and the mechanical forces generated on the organ scale. Here we present a computational model describing cellular division and migration on the cellular scale, as well as growth and cortical folding on the tissue or organ scale, in a continuous way by a coupled finite growth and advection-diffusion model. We introduce the cell density as an independent field controlling the volumetric growth. Furthermore, we formulate a positive relation between cell density and cortical layer stiffness. This allows us to study the influence of the migration velocity, the cell diffusivity, the local stiffness, and the local connectivity of cells on the cortical folding process and mechanical properties during normal and abnormal brain development numerically. We show how an increase in the density of the neurons increases the layer's mechanical stiffness. Moreover, weWe validate our simulation results through the comparison with histological sections of the fetal human brain. The current model aims to be a first step towards providing a reliable platform to systematically evaluate the role of different cellular events on the cortical folding process and vice vers

General imperfect interfaces - Javili, Ali; Käßmair, Stefan; Steinmann, Paul (Computer Methods in Applied Mechanics and Engineering, 2014)
The objective of this contribution is to develop a thermodynamically consistent theory for general imperfect interfaces and to establish a unified computational framework to model all classes of such interfaces using the finite element method. The interface is termed general imperfect in the sense that it allows for a jump of the temperature as well as for a jump of the normal heat flux across the interface. Conventionally, imperfect interfaces with respect to their thermal behavior are restricted to being either highly-conducting (HC) or lowly-conducting (LC) also known as Kapitza. For a HC interface the temperature is continuous across the interface while the jump of the normal heat flux is admissible. On the contrary, a LC interface does not allow for a jump of the heat flux across the interface but it does allow for a temperature jump. The temperature jump of a LC interface is frequently assumed to be proportional to the average heat flux across the interface. In this contribution we prove that this common assumption is indeed an appropriate condition to (sufficiently and not necessarily) satisfy the second law of thermodynamics. While HC and LC interfaces are generally accepted and well established today, the general imperfect interfaces remain poorly understood. Here we propose a thermodynamically consistent theory of general imperfect interfaces and we show that the dissipative structure of the interface suggests firstly to classify such interfaces as semi-dissipative (SD) and fully-dissipative (FD). Secondly, for a FD interface the interface temperature shall be considered as an independent degree of freedom and a new (constitutive) equation is obtained to calculate the interface temperature using a new interface material parameter i.e. the sensitivity. Furthermore, we show how all types of interfaces are derived from a FD general imperfect interface model. This finding allows us to establish a unified finite element framework to model all classes of interfaces. Full details of the novel numerical scheme are provided. Key features of general imperfect interfaces are then elucidated via a series of three-dimensional numerical examples. In particular, we show that according to the second law the interface temperature may not necessarily be the average of (or even between) the temperatures across the interface. Finally, we recall since the influence of interfaces on the overall response of a body increases as the scale of the problem decreases, this contribution has certain applications to nano-composites.
UnivIS-Import:2015-03-09:Pub.2013.tech.FT.FT-TM.genera
Keywords: General imperfect interface; Highly-conducting; Lowly-conducting; Kapitza; Nano-materials; Finite element method

Inverse identification of region-specific hyperelastic material parameters for human brain tissue - Hinrichsen, Jan; Reiter, Nina; Bräuer, Lars; Paulsen, Friedrich; Käßmair, Stefan; Budday, Silvia (Biomechanics and Modeling in Mechanobiology, 2023)
The identification of material parameters accurately describing the region-dependent mechanical behavior of human brain tissue is crucial for computational models used to assist, e.g., the development of safety equipment like helmets or the planning and execution of brain surgery. While the division of the human brain into different anatomical regions is well established, knowledge about regions with distinct mechanical properties remains limited. Here, we establish an inverse parameter identification scheme using a hyperelastic Ogden model and experimental data from multi-modal testing of tissue from 19 anatomical human brain regions to identify mechanically distinct regions and provide the corresponding material parameters. We assign the 19 anatomical regions to nine governing regions based on similar parameters and microstructures. Statistical analyses confirm differences between the regions and indicate that at least the corpus callosum and the corona radiata should be assigned different material parameters in computational models of the human brain. We provide a total of four parameter sets based on the two initial Poisson’s ratios of 0.45 and 0.49 as well as the pre- and unconditioned experimental responses, respectively. Our results highlight the close interrelation between the Poisson’s ratio and the remaining model parameters. The identified parameters will contribute to more precise computational models enabling spatially resolved predictions of the stress and strain states in human brains under complex mechanical loading conditions.<br

Inverse identification of region-specific hyperelastic material parameters for human brain tissue - Hinrichsen, Jan; Reiter, Nina; Bräuer, Lars; Paulsen, Friedrich; Käßmair, Stefan; Budday, Silvia (Biomechanics and modeling in mechanobiology, 2023)
The identification of material parameters accurately describing the region-dependent mechanical behavior of human brain tissue is crucial for computational models used to assist, e.g., the development of safety equipment like helmets or the planning and execution of brain surgery. While the division of the human brain into different anatomical regions is well established, knowledge about regions with distinct mechanical properties remains limited. Here, we establish an inverse parameter identification scheme using a hyperelastic Ogden model and experimental data from multi-modal testing of tissue from 19 anatomical human brain regions to identify mechanically distinct regions and provide the corresponding material parameters. We assign the 19 anatomical regions to nine governing regions based on similar parameters and microstructures. Statistical analyses confirm differences between the regions and indicate that at least the corpus callosum and the corona radiata should be assigned different material parameters in computational models of the human brain. We provide a total of four parameter sets based on the two initial Poisson’s ratios of 0.45 and 0.49 as well as the pre- and unconditioned experimental responses, respectively. Our results highlight the close interrelation between the Poisson’s ratio and the remaining model parameters. The identified parameters will contribute to more precise computational models enabling spatially resolved predictions of the stress and strain states in human brains under complex mechanical loading conditions.
CRIS-Team Scopus Importer:2023-09-15
Keywords: Finite element method; Finite hyperelasticity; Human brain tissue; Ogden model; Parameter identification

Magnetic force and torque on particles subject to a magnetic field - Vogel, Franziska Anna; Pelteret, Jean-Paul; Käßmair, Stefan; Steinmann, Paul (European Journal of Mechanics A-Solids, 2014)
Materials that are sensitive to an applied magnetic field are of increased interest and use to industry and researchers. The realignment of magnetizable particles embedded within a substrate results in a deformation of the material and alteration of its intrinsic properties. An increased understanding of the influence of the particles under magnetic load is required to better predict the behaviour of the material. In this work, we examine two distinct approaches to determine the resulting magnetic force and torque generated within a general domain. The two methodologies are qualitatively and quantitatively compared, and we propose scenarios under which one is more suitable for use than the other. We also describe a method to compute the generated magnetic torque. These post-processing procedures utilize results derived from a magnetic scalar-potential formulation for the large deformation magneto-elastic problem. We demonstrate their application in several examples involving a single and two particle system embedded within a carrier matrix. It is shown that, given a chosen set of boundary conditions, the magnetic forces and torques acting on a particle are influenced by its shape, size and location within the carrier.
UnivIS-Import:2015-03-09:Pub.2013.tech.FT.FT-TM.magnet
Keywords: Magnetoactive materials; Magnetoelasticity; Finite-element method

Physical aspects of cortical folding - Greiner, Alexander; Käßmair, Stefan; Budday, Silvia (Soft Matter, 2021)
The characteristically folded surface of the human brain is critical for brain function and allows for higher cognitive abilities. Recent mostly computational research advances have shown that mechanical instabilities play a crucial role during early brain development and cortical folding. However, it is difficult to investigate such mechanisms in vivo. To experimentally gain deeper insights into the physical mechanisms that underlie the development of brain shape, we use a setup of swelling polymers. We investigate the influence of cortical thickness and the stiffness ratio between cortex and subcortex on the resulting surface pattern by taking the initially smooth fetal brain geometry at week 22 into consideration. The gel specimens possess a two-layered structure accounting for gray and white matter tissue and yield complex surface morphologies that well resemble patterns in the human brain. The results are in good agreement with analytical predictions. Through the variation of cortical thickness and stiffness, it is possible to reproduce cortical malformations such as polymicrogyria and lissencephaly. The results suggest that wrinkling with subsequent transition into folding is the driving instability mechanism during brain development. In addition, the experiments provide valuable insights towards the distinction between wrinkling and creasing instabilities. Taken together, the presented swelling experiments impressively demonstrate the purely physical aspects of brain shape and constitute a valuable tool to advance our understanding of human brain development.

Thermomechanics of solids with general imperfect coherent interfaces - Käßmair, Stefan; Javili, Ali; Steinmann, Paul (Archive of Applied Mechanics, 2014)
The objective of this contribution is to develop a thermodynamically consistent theory for general imperfect coherent interfaces in view of their thermomechanical behavior and to establish a unified computational framework to model all classes of such interfaces using the finite element method. Conventionally, imperfect interfaces with respect to their thermal behavior are often restricted to being either highly conducting (HC) or lowly conducting (LC) also known as Kapitza. The interface model here is general imperfect in the sense that it allows for a jump of the temperature as well as for a jump of the normal heat flux across the interface. Clearly, in extreme cases, the current model simplifies to HC and LC interfaces. A new characteristic of the general imperfect interface is that the interface temperature is an independent degree of freedom and, in general, is not a function of only temperatures across the interface. The interface temperature, however, must be computed using a new interface material parameter, i.e., the sensitivity. It is shown that according to the second law, the interface temperature may not necessarily be the average of (or even between) the temperatures across the interface. In particular, even if the temperature jump at the interface vanishes, the interface temperature may be different from the temperatures across the interface. This finding allows for a better, and somewhat novel, understanding of HC interfaces. That is, a HC interface implies, but is not implied by, the vanishing temperature jump across the interface. The problem is formulated such that all types of interfaces are derived from a general imperfect interface model, and therefore, we establish a unified finite element framework to model all classes of interfaces for general transient problems. Full details of the novel numerical scheme are provided. Key features of the problem are then elucidated via a series of three-dimensional numerical examples. Finally, we recall since the influence of interfaces on the overall response of a body increases as the scale of the problem decreases, this contribution has certain applications to nano-composites and also thermal interface materials.
CRIS-Team Scopus Importer:2022-08-05
Keywords: Finite element method; General imperfect interface; Generalized Young-Laplace; Highly conducting; Kapitza; Lowly conducting; Nano-materials; Thermal interface materials

Variationally consistent computational homogenization of chemomechanical problems with stabilized weakly periodic boundary conditions - Käßmair, Stefan; Runesson, Kenneth; Steinmann, Paul; Jänicke, Ralf; Larsson, Fredrik (International Journal for Numerical Methods in Engineering, 2021)
A variationally consistent model-based computational homogenization approach for transient chemomechanically coupled problems is developed based on the classical assumption of first-order prolongation of the displacement, chemical potential, and (ion) concentration fields within a representative volume element (RVE). The presence of the chemical potential and the concentration as primary global fields represents a mixed formulation, which has definite advantages. Nonstandard diffusion, governed by a Cahn–Hilliard type of gradient model, is considered under the restriction of miscibility. Weakly periodic boundary conditions on the pertinent fields provide the general variational setting for the uniquely solvable RVE-problem(s). These boundary conditions are introduced with a novel approach in order to control the stability of the boundary discretization, thereby circumventing the need to satisfy the LBB-condition: the penalty stabilized Lagrange multiplier formulation, which enforces stability at the cost of an additional Lagrange multiplier for each weakly periodic field (three fields for the current problem). In particular, a neat result is that the classical Neumann boundary condition is obtained when the penalty becomes very large. In the numerical examples, we investigate the following characteristics: the mesh convergence for different boundary approximations, the sensitivity for the choice of penalty parameter, and the influence of RVE-size on the macroscopic response.
CRIS-Team Scopus Importer:2021-08-20
Keywords: chemomechanical coupling; computational homogenization; Dirichlet and Neumann RVE-conditions; failsafe; variationally consistent; weak periodicity