Multiscale Problems Research Articles

The role of flow in the self-assembly of dragline spider silk proteins

Hydrodynamic flow in the spider duct induces conformational changes in dragline spider silk proteins (spidroins) and drives their assembly, but the underlying physical mechanisms are still elusive. Here we address this challenging multiscale problem with a complementary strategy of atomistic and coarse-grained molecular dynamics simulations with uniform flow. The conformational changes at the molecular level were analyzed for single-tethered spider silk peptides. Uniform flow leads to coiled-to-stretch transitions and pushes alanine residues into β sheet and poly-proline II conformations. Coarse-grained simulations of the assembly process of multiple semi-flexible block copolymers using multi-particle collision dynamics reveal that the spidroins aggregate faster but into low-order assemblies when they are less extended. At medium-to-large peptide extensions (50%–80%), assembly slows down and becomes reversible with frequent association and dissociation events, whereas spidroin alignment increases and alanine repeats form ordered regions. Our work highlights the role of flow in guiding silk self-assembly into tough fibers by enhancing alignment and kinetic reversibility, a mechanism likely relevant also for other proteins whose function depends on hydrodynamic flow.

A physics-driven Σ-Y atomization model for heavy-duty engine simulations

The atomization of a liquid jet is a multi-scale and multi-physics problem of interest for many engineering applications. Particularly, it drives the fuel-oxidizer mixing and dictates the efficiency of combustion engines. High-fidelity multi-phase simulations remain challenging due to the excessive computational cost required to capture all the atomization scales. Thus, atomization models are necessary to represent sub-grid liquid structures. In this work, a modified Σ-Y model in the context of Eulerian-Lagrangian Spray Atomization (ELSA) is used to transport the surface area density of the spray. The model's predictive performance is assessed under various operating conditions relevant to heavy-duty engines using the Engine Combustion Network (ECN) Spray C and Spray D research-grade injectors. The classic droplet collision formulation of the Σ-Y model alone does not replicate the response of the measured spray surface area to changes in injector, ambient pressure, and injection pressure, requiring individual tuning of the model's parameters. Instead, a transition between dense and dilute spray breakup mechanisms is proposed in terms of the average droplet spacing. The collision breakup mechanism represents the dilute spray, whereas the droplet size in the dense spray is driven by a competition between the integral scale of turbulence and the balance between the turbulent kinetic energy and the surface energy of the droplets. Such an approach minimizes the requirements for model tuning. Moreover, the role of the constants of a compressible standard k-ε RANS model is assessed in the injector's internal flow and external spray simulation framework, and an updated set is proposed. The results are validated against x-ray radiography and Ultra-Small Angle X-ray Scattering (USAXS) data and highlight the predictive capabilities of the proposed physics-driven Σ-Y model, which is compatible with engine simulation turnaround times.

A variational approach for temporal multiscale problems and its application to adaptivity and optimization

AbstractWe discuss a computational multiscale method for the efficient simulation and optimization of differential equations with multiple scales in time. The approach is based on isolating the influences of the fast scale by a local periodic‐in‐time assumption that allows for time‐stepping of the averaged slow scale with very large step sizes. For this approach we develop a variational formulation that gives easy access to the methodology of goal‐oriented error estimation and adaptivity as well as gradient based optimization.

Optimization of structural parameters and numerical simulation of stress field of composite crucible based on the indirect coupling method

Abstract The research starts with the treatment of the multiscale transmission problem and establishes the electromagnetic solidification transmission coupling mathematical model based on the indirect coupling method. It uses the three-dimensional magnetic field finite element theory to establish a three-dimensional crucible structure continuous casting model built on the electromagnetic solidification transmission coupling mathematical model. This model is used to optimize the parameters of the composite crucible structure and to simulate electromagnetic transmission and braking phenomena. The results show that the L-shaped static magnetic field has a more potent inhibition and a guidance effect on melt circulation. The braking effect of the actual magnetic field on the downward impact is worse. Under the influence of an L-shaped magnetic field, the flow velocity of the melt is better, and the flow state distribution is more smooth and uniform. The computational efficiency test results show that the conversion calculation time of the method designed in this study is 18.03 min. The total calculation time is 680.48 min, which is superior to traditional methods. It proves that this model can accurately analyze the magnetic field coupling problem and at the same time ensure the superiority of its computing efficiency.

Multiscale and multi-physical problems: Comment on "What is life? Active particles tools towards behavioral dynamics in social-biology and economics" by N. Bellomo, M. Esfahanian, V. Secchini, and P. Terna.

Multiscale and multi-physical problems: Comment on "What is life? Active particles tools towards behavioral dynamics in social-biology and economics" by N. Bellomo, M. Esfahanian, V. Secchini, and P. Terna.

Analysis of Radar Target Scattering Echo With Surface Ducting in Large-Scale Environments Based on the PE-MoM Hybrid Method

This letter proposes a three-dimensional (3-D) PE- MoM hybrid method to predict radar target scattering with sur- face ducting in large-scale environments. The method divides the complex multiscale problem into two sub-problems: electromag- netic (EM) wave propagation and scattering. The bidirectional 3-D parabolic equation (PE) method is used to consider the influence of environmental factors such as irregular terrain and atmospheric duct on EM wave propagation, while the method of moments (MoM) is used to calculate the scattering from the key target. By introducing phase compensation and Fourier synthesis techniques, the proposed method can be extended to predict transient scattering echoes from the target. A typical example is given to verify the correctness of the proposed method, and good consistency is also seen from comparison with the MoM. Finally, simulation analysis is carried out for the scattering of a missile target in a sea-land composite environment. The results show that the proposed method can effectively predict the scattering echoes of low-altitude targets in large-scale environments.

Solving multiscale elliptic problems by sparse radial basis function neural networks

Machine learning has been successfully applied to various fields of scientific computing in recent years. For problems with multiscale features such as flows in porous media and mechanical properties of composite materials, however, it remains difficult for machine learning methods to resolve such multiscale features, especially when the small scale information is present. In this work, we propose a sparse radial basis function neural network method to solve elliptic partial differential equations (PDEs) with multiscale coefficients. Inspired by the deep mixed residual method, we rewrite the second-order problem into a first-order system and employ multiple radial basis function neural networks (RBFNNs) to approximate unknown functions in the system. To avoid the overfitting and improve the performance of RBFNN, an additional regularization is introduced in the loss function. Thus the loss function contains two parts: the L2 loss for the residual of the first-order system and boundary conditions, and the ℓ1 regularization term for the weights of radial basis functions (RBFs). An algorithm for optimizing the specific loss function is introduced to accelerate the training process. The accuracy and effectiveness of the proposed method are demonstrated through a collection of multiscale problems with scale separation, discontinuity and multiple scales from one to three dimensions. Notably, the ℓ1 regularization can achieve the goal of representing the solution by fewer RBFs. As a consequence, the total number of RBFs scales like O(ε−nτ), where ε is the smallest scale, n is the dimensionality, and τ is typically smaller than 1. It is worth mentioning that the proposed method not only has the numerical convergence and thus provides a reliable numerical solution in three dimensions when a classical method is typically not affordable, but also outperforms most other available machine learning methods in terms of accuracy and robustness. Executable files are available at https://github.com/wangzhiwensuda/SRBFNN-for-multiscale-elliptic-problems. The program was implemented in PyTorch and was run on Linux.

Thermodynamically consistent concurrent material and structure optimization of elastoplastic multiphase hierarchical systems.

The concept of concurrent material and structure optimization aims at alleviating the computational discovery of optimum microstructure configurations in multiphase hierarchical systems, whose macroscale behavior is governed by their microstructure composition that can evolve over multiple length scales from a few micrometers to centimeters. It is based on the split of the multiscale optimization problem into two nested sub-problems, one at the macroscale (structure) and the other at the microscales (material). In this paper, we establish a novel formulation of concurrent material and structure optimization for multiphase hierarchical systems with elastoplastic constituents at the material scales. Exploiting the thermomechanical foundations of elastoplasticity, we reformulate the material optimization problem based on the maximum plastic dissipation principle such that it assumes the format of an elastoplastic constitutive law and can be efficiently solved via modified return mapping algorithms. We integrate continuum micromechanics based estimates of the stiffness and the yield criterion into the formulation, which opens the door to a computationally feasible treatment of the material optimization problem. To demonstrate the accuracy and robustness of our framework, we define new benchmark tests with several material scales that, for the first time, become computationally feasible. We argue that our formulation naturally extends to multiscale optimization under further path-dependent effects such as viscoplasticity or multiscale fracture and damage.

Multiscale and multi-physical problems: Comment to “Mechanotransduction in tumor dynamics modeling”, by B. Blanco, H. Gomezc, J. Melchor, R. Palma, J. Soler, and G. Rus

Multiscale and multi-physical problems: Comment to “Mechanotransduction in tumor dynamics modeling”, by B. Blanco, H. Gomezc, J. Melchor, R. Palma, J. Soler, and G. Rus

Enabling supra-aortic vessels inclusion in statistical shape models of the aorta: a novel non-rigid registration method.

Statistical Shape Models (SSMs) are well-established tools for assessing the variability of 3D geometry and for broadening a limited set of shapes. They are widely used in medical imaging due to their ability to model complex geometries and their high efficiency as generative models. The principal step behind these techniques is a registration phase, which, in the case of complex geometries, can be a critical issue due to the correspondence problem, as it necessitates the development of correspondence mapping between shapes. The thoracic aorta, with its high level of morphological complexity, poses a multi-scale deformation problem due to the presence of several branch vessels with varying diameters. Moreover, branch vessels exhibit significant variability in shape, making the correspondence optimization even more challenging. Consequently, existing studies have focused on developing SSMs based only on the main body of the aorta, excluding the supra-aortic vessels from the analysis. In this work, we present a novel non-rigid registration algorithm based on optimizing a differentiable distance function through a modified gradient descent approach. This strategy enables the inclusion of custom, domain-specific constraints in the objective function, which act as landmarks during the registration phase. The algorithm's registration performance was tested and compared to an alternative Statistical Shape modeling framework, and subsequently used for the development of a comprehensive SSM of the thoracic aorta, including the supra-aortic vessels. The developed SSM was further evaluated against the alternative framework in terms of generalisation, specificity, and compactness to assess its effectiveness.

Mathematical models of developmental vascular remodelling: A review.

Over the past 40 years, there has been a strong focus on the development of mathematical models of angiogenesis, while developmental remodelling has received little such attention from the mathematical community. Sprouting angiogenesis can be seen as a very crude way of laying out a primitive vessel network (the raw material), while remodelling (understood as pruning of redundant vessels, diameter control, and the establishment of vessel identity and hierarchy) is the key to turning that primitive network into a functional network. This multiscale problem is of prime importance in the development of a functional vasculature. In addition, defective remodelling (either during developmental remodelling or due to a reactivation of the remodelling programme caused by an injury) is associated with a significant number of diseases. In this review, we discuss existing mathematical models of developmental remodelling and explore the important contributions that these models have made to the field of vascular development. These mathematical models are effectively used to investigate and predict vascular development and are able to reproduce experimentally observable results. Moreover, these models provide a useful means of hypothesis generation and can explain the underlying mechanisms driving the observed structural and functional network development. However, developmental vascular remodelling is still a relatively new area in mathematical biology, and many biological questions remain unanswered. In this review, we present the existing modelling paradigms and define the key challenges for the field.

Atomistic and continuum length scale coupling in materials using quasicontinuum method

We propose coupling atomistic and continuum length scale models in materials to develop a multiscale modeling framework. High fidelity models like Molecular Dynamics and Density Functional Theory exist at atomic scales capable of explaining various small-scale occurrences. However, these models cannot be scaled up efficiently to explain a range of observations occurring at multiple length and time scales. Since continuum properties can be considered emergent from numerous smaller-scale phenomena, models capable of effectively relating multiple length and time scales are needed. Therefore, we propose using existing models at differing yet individual length scales and suitably coupling them for tackling such multiscale problems. This coupling method should facilitate information exchange in real-time and demonstrate rapid convergence without adding significant computational overhead. Together with single-scale models, this coupling could help us relate continuum properties like stiffness, damping, and thermal conductivity to a material's electronic structure and atomic arrangement. As an illustration, we choose a simple one-dimensional problem with a limited region of interest which we wish to study at an atomic scale. We rely upon the Quasicontinuum method to model this limited atomic region, which significantly reduces the degrees of freedom involved for the atomistic subdomain. Preliminary results for this static problem indicate that the displacement error is relevant on the atomic scale. Importance is put on deriving a method capable of effectively coupling atomic and continuum subdomains for solving general static or dynamic structural problems.

BelNet: basis enhanced learning, a mesh-free neural operator

Operator learning trains a neural network to map functions to functions. An ideal operator learning framework should be mesh-free in the sense that the training does not require a particular choice of discretization for the input functions, allows for the input and output functions to be on different domains, and is able to have different grids between samples. We propose a mesh-free neural operator for solving parametric partial differential equations. The basis enhanced learning network (BelNet) projects the input function into a latent space and reconstructs the output functions. In particular, we construct part of the network to learn the ‘basis’ functions in the training process. This generalized the networks proposed in Chen & Chen (Chen and Chen 1995 IEEE Trans. Neural Netw. 49 , 911–917. ( doi:10.1109/72.392253 ) and 6 , 904–910. ( doi:10.1109/IJCNN.1993.716815 )) to account for differences in input and output meshes. Through several challenging high-contrast and multiscale problems, we show that our approach outperforms other operator learning methods for these tasks and allows for more freedom in the sampling and/or discretization process.

Cross-Modal Transformer for RGB-D semantic segmentation of production workshop objects

Scene understanding in a production workshop is an important technology to improve its intelligence level, semantic segmentation of production workshop objects is an effective method for realizing scene understanding. Since the varieties of information of production workshop, making full use of the complementary information of RGB image and depth image can effectively improve the semantic segmentation accuracy of production workshop objects. Aiming at solving the multi-scale and real-time problems of segmenting the production workshop objects, this paper proposes Cross-Modal Transformer (CMFormer), a Transformer-based cross-modal semantic segmentation model. Its key feature correction and feature fusion parts are composed of the Multi-Scale Channel Attention Correction(MS-CAC) module and the Global Feature Aggregation(GFA) module. By improving Multi-Head Self-Attention(MHSA) in Transformer, we design Cross-Modal Multi-Head Self-Attention(CM-MHSA) to build long-range interaction between RGB image and depth image, and further design the MS-CAC module and the GFA module on the basis of the CM-MHSA module, to achieve cross-modal information interaction in the channel and spatial dimensions. Among them, the MS-CAC module enriches the multi-scale features of each channel and achieve more accurate channel attention correction between the two modals; the GFA module interacts with RGB feature and depth feature in the spatial dimension and fuses global and local features at the same time. In the experiments on the NYU Depth v2 dataset, the CMFormer reached 68.00% MPA(Mean Pixel Accuracy) and 55.75% mIoU(Mean Intersection over Union), achieves the state-of-the-art results. While in the experiments on the Scene Objects for Production workshop dataset(SOP), the CMFormer achieves 96.74% MPA, 92.98% mIoU and 43 FPS(Frames Per Second), which has high precision and good real-time performance. Code is available at: https://github.com/FutureIAI/CMFormer

An adaptive space and time method in partially explicit splitting scheme for multiscale flow problems

We propose a space and time adaptive framework for a partially explicit splitting scheme of flow problems in multiscale media. We consider the parabolic equation with high-contrast multiscale coefficients. It is known that sufficient number of multiscale basis are needed to capture non-local features and information across all scales. Moreover, the time step size in an explicit discretization of the problem is affected by the magnitude of the coefficient. To improve computational efficiency, in this work, we introduce an adaptive algorithm to refine temporal size and enrich multiscale spaces. We first utilize a stable temporal splitting scheme with an appropriate construction of multiscale spaces to ensure that the time step size is independent of the contrast. The multiscale subspaces are then constructed to handle the fast-flow and slow-flow parts separately, and implicit and explicit discretization is adopted for the degrees of freedom with respect to the two multiscale subspaces, respectively. A multirate time stepping is introduced for the two parts considering that the flow rates vary significantly in different regions due to the high-contrast features of the multiscale coefficients. We derive both temporal and spatial error estimators to identify local regions for two components of the solutions. Start with coarser time step sizes and a few important multiscale basis functions in each subspace, we can adaptively refine time stepping and add additional basis with constraint energy minimizing properties based on the derived error indicators. The stability and convergence of the method are studied, and several numerical tests are presented to demonstrate the performance of the proposed method.

Finite basis physics-informed neural networks (FBPINNs): a scalable domain decomposition approach for solving differential equations

Recently, physics-informed neural networks (PINNs) have offered a powerful new paradigm for solving problems relating to differential equations. Compared to classical numerical methods, PINNs have several advantages, for example their ability to provide mesh-free solutions of differential equations and their ability to carry out forward and inverse modelling within the same optimisation problem. Whilst promising, a key limitation to date is that PINNs have struggled to accurately and efficiently solve problems with large domains and/or multi-scale solutions, which is crucial for their real-world application. Multiple significant and related factors contribute to this issue, including the increasing complexity of the underlying PINN optimisation problem as the problem size grows and the spectral bias of neural networks. In this work, we propose a new, scalable approach for solving large problems relating to differential equations called finite basis physics-informed neural networks (FBPINNs). FBPINNs are inspired by classical finite element methods, where the solution of the differential equation is expressed as the sum of a finite set of basis functions with compact support. In FBPINNs, neural networks are used to learn these basis functions, which are defined over small, overlapping subdomains. FBINNs are designed to address the spectral bias of neural networks by using separate input normalisation over each subdomain and reduce the complexity of the underlying optimisation problem by using many smaller neural networks in a parallel divide-and-conquer approach. Our numerical experiments show that FBPINNs are effective in solving both small and larger, multi-scale problems, outperforming standard PINNs in both accuracy and computational resources required, potentially paving the way to the application of PINNs on large, real-world problems.

DL-MSTO+: A deep learning-based multi-scale topology optimization framework via positive definiteness ensured material representation network

A multi-scale approach to topology optimization has recently emerged due to its lightweight, robust, and multi-functional characteristics. Considering material diversity, an increasing number of materials leads to a computational burden due to numerous design variables and homogenization equations. This study proposes DL-MSTO+, a deep learning-based multi-scale topology optimization framework. The framework consists of two distinct deep neural networks for multi-scale topology optimization problems to reduce the dimensionality of design variables and predict homogenized material properties. First, a generator network learns the low-dimensional representation of a material microstructure in an unsupervised manner. Thus, the trained generator produces a micro-structural image from a low-dimensional vector. Second, a predictor network is trained in a supervised manner to predict the homogenized elasticity matrix for a given micro-structural image. Additionally, the predictor is specifically designed to ensure the positive definiteness of the predicted elasticity matrix. Lastly, the networks are integrated into the algorithm to improve the efficiency of multi-scale topology optimization. The numerical experiments demonstrate higher efficiency for the proposed algorithm than the conventional multi-scale approach. Moreover, the proposed method provides connectable multi-scale designs, and the low-dimensional latent representation enables semantic interpolation between solutions.

On a procedure to derive ϵ-factorised differential equations beyond polylogarithms

In this manuscript, we elaborate on a procedure to derive ϵ-factorised differential equations for multi-scale, multi-loop classes of Feynman integrals that evaluate to special functions beyond multiple polylogarithms. We demonstrate the applicability of our approach to diverse classes of problems, by working out ϵ-factorised differential equations for single- and multi-scale problems of increasing complexity. To start we are reconsidering the well-studied equal-mass two-loop sunrise case, and move then to study other elliptic two-, three- and four-point problems depending on multiple different scales. Finally, we showcase how the same approach allows us to obtain ϵ-factorised differential equations also for Feynman integrals that involve geometries beyond a single elliptic curve.

Dilated residual FPN-based segmentation for mouse retinal images

Background and objectiveDiabetes can induce diabetic retinopathy (DR), and the blindness caused by this disease is irreversible. The early analysis of mouse retinal images, including the layer and cell segmentation properties of these images, can help to effectively diagnose this disease. MethodIn the study, we design a dilated residual method based on a feature pyramid network (FPN), in which the FPN is adopted as the base network for solving the multiscale segmentation problem concerning mouse retinal images. In the bottom-up encoding pathway, we construct our backbone feature extraction network via the combination of dilated convolution and a residual block, further increasing the range of the receptive field to obtain more context information. At the same time, we integrate a squeeze-and-excitation (SE) attention module into the backbone network to obtain more small object details. In the top-down decoding pathway, we replace the traditional nearest-neighbor upsampling method with the transposed convolution method and add a segmentation head to obtain semantic segmentation results. ResultsThe effectiveness of our network model is verified in two segmentation tasks: ganglion cell segmentation and mouse retinal cell and layer segmentation. The outcomes demonstrate that, compared to other supervised segmentation methods based on deep learning, our model attains the utmost precision in both binary segmentation and multiclass semantic segmentation tasks. ConclusionThe dilated residual FPN is a robust method for mouse retinal image segmentation and it can effectively assist DR diagnosis.

Ultrasound-mediated nano-sized drug delivery systems for cancer treatment: Multi-scale and multi-physics computational modeling.

Computational modeling enables researchers to study and understand various complex biological phenomena in anticancer drug delivery systems (DDSs), especially nano-sized DDSs (NSDDSs). The combination of NSDDSs and therapeutic ultrasound (TUS), that is, focused ultrasound and low-intensity pulsed ultrasound, has made significant progress in recent years, opening many opportunities for cancer treatment. Multiple parameters require tuning and optimization to develop effective DDSs, such as NSDDSs, in which mathematical modeling can prove advantageous. In silico computational modeling of ultrasound-responsive DDS typically involves a complex framework of acoustic interactions, heat transfer, drug release from nanoparticles, fluid flow, mass transport, and pharmacodynamic governing equations. Owing to the rapid development of computational tools, modeling the different phenomena in multi-scale complex problems involved in drug delivery to tumors has become possible. In the present study, we present an in-depth review of recent advances in the mathematical modeling of TUS-mediated DDSs for cancer treatment. A detailed discussion is also provided on applying these computational models to improve the clinical translation for applications in cancer treatment. This article is categorized under: Nanotechnology Approaches to Biology > Nanoscale Systems in Biology Therapeutic Approaches and Drug Discovery > Nanomedicine for Oncologic Disease.