Adaptive Finite Element Solver Research Articles

An adaptive semi-implicit finite element solver for brain cancer progression modeling.

Glioblastoma is the most aggressive and infiltrative glioma, classified as Grade IV, with the poorest survival rate among patients. Accurate and rigorously tested mechanistic in silico modeling offers great value to understand and quantify the progression of primary brain tumors. This paper presents a continuum-based finite element framework that is built on high performance computing, open-source libraries to simulate glioblastoma progression. We adopt the established proliferation invasion hypoxia necrosis angiogenesis model in our framework to realize scalable simulations of cancer, and has demonstrated to produce accurate and efficient solutions in both two- and three-dimensional brain models. The in silico solver can successfully implement arbitrary order discretization schemes and adaptive remeshing algorithms. A model sensitivity analysis is conducted to test the impact of vascular density, cancer cell invasiveness and aggressiveness, the phenotypic transition potential, including that of necrosis, and the effect of tumor-induced angiogenesis in the evolution of glioblastoma. Additionally, individualized simulations of brain cancer progression are carried out using pertinent magnetic resonance imaging data, where the in silico model is used to investigate the complex dynamics of the disease. We conclude by arguing how the proposed framework can deliver patient-specific simulations of cancer prognosis and how it could bridge clinical imaging with modeling.

Comparison of an [formula omitted]- and [formula omitted]-adaptive finite element solver for chemo-mechanically coupled battery electrode particles

Numerical investigations of mechanical stresses for phase transforming battery electrode materials on the particle scale are computationally highly demanding. The limitations are mainly induced by the strongly varying spatial and temporal scales of the underlying phase field model, which require an ultra fine mesh and time resolution, however, solely at specific stages in space and time. To overcome these numerical difficulties we present a general-purpose space and time adaptive solution algorithm based on an hp-adaptive finite element method and a variable-step, variable-order time integrator. At the example of a chemo-mechanical electrode particle model we demonstrate the computational savings gained by the hp-adaptivity. In particular, we compare the results to an h-adaptive finite element method and show the reduction of computational complexity.

An open source [formula omitted]-adaptive discontinuous Galerkin finite element solver for linear elasticity

• High accuracy discontinuous Galerkin finite element code. • Automatic hp-adaptivity. • The code is easy to modify to solver other physical equations. • Easy to use Matlab code. Open source codes are a key ingredient to greater research integrity and accountability in computational science and engineering. However, many of these codes have not been developed with modification of the base code as their primary consideration. Existing codes may provide an environment for researchers to quickly test out their ideas under different physical conditions in a high level way but they are not always ideal for those interested in the development of numerical methods. The majority of existing open source discontinuous Galerkin finite element codes are written in C++ and there is a significant learning curve for junior researchers to adopt, understand and modify the underlying code/routines. This paper presents an open source h p -adaptive discontinuous Galerkin finite element code written in MATLAB that has been explicitly designed to make it easy for users, especially MSc/PhD-level researchers, to understand the method and implement new ideas within the core code. Although the code is focused on solving problems in linear elasticity, it is straightforward to modify it to solve other physical equations.

ARGOS: An adaptive refinement goal-oriented solver for the linearized Poisson-Boltzmann equation.

An adaptive finite element solver for the numerical calculation of the electrostatic coupling between molecules in a solvent environment is developed and tested. At the heart of the solver is a goal-oriented a posteriori error estimate for the electrostatic coupling, derived and implemented in the present work, that gives rise to an orders of magnitude improved precision and a shorter computational time as compared to standard finite difference solvers. The accuracy of the new solver ARGOS is evaluated by numerical experiments on a series of problems with analytically known solutions. In addition, the solver is used to calculate electrostatic couplings between two chromophores, linked to polyproline helices of different lengths and between the spike protein of SARS-CoV-2 and the ACE2 receptor. All the calculations are repeated by using the well-known finite difference solvers MEAD and APBS, revealing the advantages of the present finite element solver.

Simple technique for unstructured mesh generation via adaptive finite elements

This work describes a concise algorithm for the generation of triangular meshes with the help of standard adaptive finite element methods. We demonstrate that a generic adaptive finite element solver can be repurposed into a triangular mesh generator if a robust mesh smoothing algorithm is applied between the mesh refinement steps. We present an implementation of the mesh generator and demonstrate the resulting meshes via examples.

Instabilities driven by frontal polymerization in thermosetting polymers and composites

Frontal Polymerization (FP) was recently demonstrated as a faster, more energy-efficient way to manufacture fiber-reinforced thermosetting-polymer-matrix composites. FP uses heat from the exothermic reaction of the solution of monomer and initiator to generate a self-propagating polymerization front. In most cases, the polymerization front propagates in a steady fashion. However, under some conditions, the front experiences instabilities, which do affect the quality of the manufactured composite part. In this work, we use a coupled thermo-chemical model and an adaptive nonlinear finite element solver to simulate FP-driven instabilities in dicyclopentadiene (DCPD) and in carbon-fiber DCPD-matrix composites. With the aid of 1-D transient simulations, we investigate how the initial temperature and the carbon fiber volume fraction affect the amplitude and wavelength of the thermal instabilities. We also extract the range of processing conditions for which the instabilities are predicted to appear. The last part of this work investigates the effect of convective heat loss on the FP-driven instabilities in both neat resin and composite cases.

An adaptive finite element procedure for fully-coupled point contact elastohydrodynamic lubrication problems

This paper presents an automatic locally adaptive finite element solver for the fully-coupled EHL point contact problems. The proposed algorithm uses a posteriori error estimation in the stress in order to control adaptivity in both the elasticity and lubrication domains. The implementation is based on the fact that the solution of the linear elasticity equation exhibits large variations close to the fluid domain on which the Reynolds equation is solved. Thus the local refinement in such region not only improves the accuracy of the elastic deformation solution significantly but also yield an improved accuracy in the pressure profile due to increase in the spatial resolution of fluid domain. Thus, the improved traction boundary conditions lead to even better approximation of the elastic deformation. Hence, a simple and an effective way to develop an adaptive procedure for the fully-coupled EHL problem is to apply the local refinement to the linear elasticity mesh. The proposed algorithm also seeks to improve the quality of refined meshes to ensure the best overall accuracy. It is shown that the adaptive procedure effectively refines the elements in the region(s) showing the largest local error in their solution, and reduces the overall error with optimal computational cost for a variety of EHL cases. Specifically, the computational cost of proposed adaptive algorithm is shown to be linear with respect to problem size as the number of refinement levels grows.

Changes in scalp potentials and spatial smoothing effects of inclusion of dura layer in human head models for EEG simulations.

The dura layer which covers the brain is less conductive than the CSF (cerebrospinal fluid) and also more conductive than the skull bone. This could significantly influence the flow of volume currents from cortex to the scalp surface which will also change the magnitude and spatial profiles of scalp potentials. This was examined with a 3-D finite element method (FEM) model of an adult subject constructed from 192 segmented axial magnetic resonance (MR) slices with 256×256 pixel resolution. The voxel resolution was 1×1×1 mm. The model included the dura layer. In addition, other major tissues were also identified. The electrical conductivities of various tissues were obtained from the literature. The conductivities of dura and CSF were 0.001 S/m and 0.06 S/m, respectively. The electrical activity of the cortex was represented by 144,000 distributed dipolar sources with orientations normal to the local cortical surface. The dipolar intensity was in the range of 0.0–0.4 mA meter with a uniform random distribution. Scalp potentials were simulated for two head models with an adaptive finite element solver. One model had the dura layer and in the other model, dura layer was replaced with the CSF. Spatial contour plots of potentials on the cortical surface, dural surface and the scalp surface were made. With the inclusion of the dura layer, scalp potentials decrease by about 20%. The contours of gyri and sulci structures were visible in the spatial profiles of the cortical potentials which were smoothed out on the dural surface and were not visible on the scalp surface. These results suggest that dura layer should be included for an accurate modeling of scalp and cortical potentials.

A parallel finite element simulator for ion transport through three‐dimensional ion channel systems

A parallel finite element simulator, ichannel, is developed for ion transport through three-dimensional ion channel systems that consist of protein and membrane. The coordinates of heavy atoms of the protein are taken from the Protein Data Bank and the membrane is represented as a slab. The simulator contains two components: a parallel adaptive finite element solver for a set of Poisson-Nernst-Planck (PNP) equations that describe the electrodiffusion process of ion transport, and a mesh generation tool chain for ion channel systems, which is an essential component for the finite element computations. The finite element method has advantages in modeling irregular geometries and complex boundary conditions. We have built a tool chain to get the surface and volume mesh for ion channel systems, which consists of a set of mesh generation tools. The adaptive finite element solver in our simulator is implemented using the parallel adaptive finite element package Parallel Hierarchical Grid (PHG) developed by one of the authors, which provides the capability of doing large scale parallel computations with high parallel efficiency and the flexibility of choosing high order elements to achieve high order accuracy. The simulator is applied to a real transmembrane protein, the gramicidin A (gA) channel protein, to calculate the electrostatic potential, ion concentrations and I - V curve, with which both primitive and transformed PNP equations are studied and their numerical performances are compared. To further validate the method, we also apply the simulator to two other ion channel systems, the voltage dependent anion channel (VDAC) and α-Hemolysin (α-HL). The simulation results agree well with Brownian dynamics (BD) simulation results and experimental results. Moreover, because ionic finite size effects can be included in PNP model now, we also perform simulations using a size-modified PNP (SMPNP) model on VDAC and α-HL. It is shown that the size effects in SMPNP can effectively lead to reduced current in the channel, and the results are closer to BD simulation results.

An Adaptive FEM for Buckling Analysis of Nonuniform Bernoulli-Euler Members via the Element Energy Projection Technique

This paper presents a new adaptive finite element (FE) procedure for accurate, efficient, and reliable computation of the critical buckling loads and the associated modes of nonuniform Bernoulli-Euler members. After the conventional FE solution on a given mesh has been obtained, a novel conceptual and practical strategy is introduced, in which the FE solution of the eigenproblem is equivalently viewed as the FE solution of an associated linear problem. This strategy allows the recently developed element energy projection (EEP) technique to be readily used to calculate super-convergent FE solutions for buckling modes, which are subsequently used to estimate the FE solution errors and to further guide mesh refinements, until the accuracy of the FE solution matches the user-preset error tolerance. Numerical examples are given to show the effectiveness, efficiency, accuracy, and reliability of the proposed method, which also paves the way for development of an adaptive FE solver for more general and complicated eigenproblems.

A parallel adaptive finite element simplified spherical harmonics approximation solver for frequency domain fluorescence molecular imaging

Fluorescence molecular imaging/tomography may play an important future role in preclinical research and clinical diagnostics. Time- and frequency-domain fluorescence imaging can acquire more measurement information than the continuous wave (CW) counterpart, improving the image quality of fluorescence molecular tomography. Although diffusion approximation (DA) theory has been extensively applied in optical molecular imaging, high-order photon migration models need to be further investigated to match quantitation provided by nuclear imaging. In this paper, a frequency-domain parallel adaptive finite element solver is developed with simplified spherical harmonics (SPN) approximations. To fully evaluate the performance of the SPN approximations, a fast time-resolved tetrahedron-based Monte Carlo fluorescence simulator suitable for complex heterogeneous geometries is developed using a convolution strategy to realize the simulation of the fluorescence excitation and emission. The validation results show that high-order SPN can effectively correct the modeling errors of the diffusion equation, especially when the tissues have high absorption characteristics or when high modulation frequency measurements are used. Furthermore, the parallel adaptive mesh evolution strategy improves the modeling precision and the simulation speed significantly on a realistic digital mouse phantom. This solver is a promising platform for fluorescence molecular tomography using high-order approximations to the radiative transfer equation.

Scalp potentials and magnetic fields of a fully and partially activated central sulcus

Abstract Our objective was to examine how the scalp potentials and the magnetic fields of a sulcus change if the cortex in the walls is partially or fully active. The scalp potentials and magnetic fields of a part of the central sulcus were simulated with a finite element model of the head constructed from the segmented MRI images of an adult male subject. The electrical activity was modeled with current dipoles (0.1 mA meter) oriented perpendicular to the curved cortical surface. The fields of scalp EEG were computed using an adaptive finite element solver. The magnetic fields at a surface 1.0 cm above the scalp were computed using Biot-Savart law. Three distributions of activity were studied: the total sulcus; the superficial half and the deep half. Contour plots of EEG and MEG were constructed for each case. The spatial patterns of the scalp potential contours were significantly different for the partial vs. full activation. Each partial activation gave a uniquely different configuration of peak MEG and EEG locations that could have diagnostic utility. The spatial shifts in the peaks were more pronounced for the MEG compared with the EEG. These results indicate that scalp potentials and fields change significantly when the electrical and magnetic manifestations of activity are selective in the walls of a sulcus.

APPLICATIONS OF CONTINUOUS SENSITIVITY EQUATIONS TO FLOWS WITH TEMPERATURE-DEPENDENT PROPERTIES

An adaptive finite-element solver is used to perform sensitivity analysis for free convection in corn syrup, a fluid with strong temperature dependence of viscosity, conductivity, and specific heat. Results are presented for sensitivity of the flow with respect to parameters describing boundary conditions, the geometry of the domain, and constitutive equations for the physical properties. This model is used to emphasize three applications of sensitivity analysis: its role in identifying key parameters controlling the flow, its use in fast calculation of nearby flows, and its utility in uncertainty analysis.

The effect of volume currents due to myocardial anisotropy on body surface potentials

Changes in anterior and posterior body surface potential maps (BSPMs) due to myocardial anisotropy were examined using a highly heterogeneous finite element model of an adult male subject constructed from segmented magnetic resonance images. A total of 23 different tissue types were identified in the whole torso. The myocardial fibre orientations in the human heart wall were mapped from the fibre orientations of a canine heart which are available in the literature using deformable mapping techniques. The current and potential distributions in the whole torso were computed using dipolar sources in the septum, apical area, left ventricular wall or right ventricular wall. For each dipole x, y, z orientations were studied. An adaptive finite element solver was used to compute currents and potential distributions in the whole torso with an element size of 0.78 × 0.78 × 3 mm in the myocardium and larger elements in other parts of the torso. For each dipole position two cases were studied. In one case the myocardium was isotropic and in the other it was anisotropic. It was found that BSPMs showed a very notable difference between the isotropic and the anisotropic myocardium for all dipole positions with the largest difference for the apical dipoles. The correlation coefficients for the BSPMs between the isotropic and anisotropic cases ranged from 0.83 for an apical dipole to 0.99 for an RV wall dipole. These results suggest that myocardial fibre anisotropy plays an important role in determining the body surface potentials.

Effect of myocardial anisotropy on the torso current flow patterns, potentials and magnetic fields*

The effects of myocardial anisotropy on the torso current flow patterns, voltage and the magnetic field were examined using an anatomically realistic torso model of an adult male subject. A finite element model of the torso was built with 19 major tissue types identified. The myocardial fibre orientation in the heart wall was included with a voxel resolution of 0.078×0.078×0.3 cm. The fibre orientations from the canine heart which are available in the literature were mapped to our adult male subject's human heart using deformable mapping techniques. The current and potential distribution in the whole torso were computed using an idealized dipolar source of ±1.0 V in the middle of the septum of the heart wall as a boundary condition. An adaptive finite element solver was used. Two cases were studied. In one case the myocardium was isotropic and in the other it was anisotropic. It was found that the current density distribution shows a very noticeable difference between the isotropic and anisotropic myocardium. The resultant magnetic field in front of the torso was computed using the Biot-Savart law. It was found that the magnetic field profile was slightly affected by the myocardial anisotropy. The potential on the torso surface also shows noticeable changes due to the myocardial anisotropy.

Atmospheric and ocean modeling with an adaptive finite element solver for the shallow-water equations

A new computational method for solving the shallow-water equations has been developed. This method combines some of the recent developments of geophysical and computational fluid dynamics, like a semi-Lagrangian advection scheme, an adaptive mesh refinement based on a posteriori error estimation, parallelization on a local memory parallel computer, and iterative solution of the linear system derived from a finite element discretization. Dynamic load balancing has been applied in order to achieve high parallel efficiency.

H-p adaptive numerical integration techniques for finite element analysis in magnetics

Advanced numerical integration methods for finite element magnetics are investigated. Standard h-type, p-type and h-p adaptive finite element solver methodologies are extended to develop analogous adaptive integration procedures. The schemes are analyzed and compared in terms of efficiency and reliability. For practical implementations, it is established that only combined h-p approaches can be guaranteed to yield reliable results, which are converged to within a pre-specified tolerance, at a reasonable computational cost. A relatively simple h-p adaptive integration method is proposed for practical magnetics applications in which accuracy and reliability are essential. In addition, the similarities and subtle differences between corresponding adaptive solver and adaptive integration approaches are identified and discussed.

An adaptive finite element solver for transient problems with moving bodies

The combination of adaptive remeshing techniques, flow solvers for transient problems with moving grids, and integrators for rigid body motion is presented. The resulting scheme allows the economic simulation of fully coupled fluid-rigid body interaction problems of arbitrary geometric complexity. Several results are given to demonstrate the capabilities developed.