Locking-Proof Tetrahedra

The simulation of incompressible materials suffers from locking when using the standard finite element method (FEM) and coarse linear tetrahedral meshes. Locking increases as the Poisson ratio gets close to 0.5 and often lower Poisson ratio values are used to reduce locking, affecting volume preservation. We propose a novel mixed FEM approach to simulating incompressible solids that alleviates the locking problem for tetrahedra. Our method uses linear shape functions for both displacements and pressure, and adds one scalar per node. It can accommodate nonlinear isotropic materials described by a Young’s modulus and any Poisson ratio value by enforcing a volumetric constitutive law. The most realistic such material is Neo-Hookean, and we focus on adapting it to our method. For , we can obtain full volume preservation up to any desired numerical accuracy. We show that standard Neo-Hookean simulations using tetrahedra are often locking, which, in turn, affects accuracy. We show that our method gives better results and that our Newton solver is more robust. As an alternative, we propose a dual ascent solver that is simple and has a good convergence rate. We validate these results using numerical experiments and quantitative analysis.

Simple numerical strategies to model freezing in variably-saturated soil with the standard finite element method

<p>An accurate representation of freezing and thawing in soil covers many applications including simulation of land surface processes, hydrology, and degrading permafrost. Freezing and thawing tightly couple water and heat flow, where temperature and temperature gradients influence the water flow and phase changes, and water content and flow influence the heat transport. In most porous media, the interface between liquid and frozen water is not sharp and a slushy zone is present. A common observation of freezing soil is water accumulation towards the freezing front due to Cryosuction. A mathematical model can be derived using the Clausius-Clapeyron equation, which allows the derivation of a soil freezing curve relating temperature to pressure head. This is based on the assumption that soil freezing is similar to soil drying.</p><p>Many models still lack features such as Cryosuction. We believe that this may be due to numerical issues that model developers face with their current solver and discretization setup. Implementing freezing soil accurately is not straight-forward. Using the Clausius-Clapeyron creates a discontinuity in the freezing rate and latent heat at the freezing point and little attention has been paid to the adequate description of their numerical treatment and computational challenges. Discretizing this discontinuous system with standard finite element methods (standard Galerkin type) can cause spurious oscillations because the standard finite element method uses continuous base/shape functions that are incapable of handling discontinuity of any kind within an element. Similarly, standard finite difference methods are also not capable of handling discontinuities. In this contribution, we present the application of regularization of the discontinuous term, which allows the use of the standard finite element method. We implemented the model in the open-source code base DRUtES (www.drutes.org). We verify this approach on synthetic and various real freezing soil column experiments conducted by Jame (1977) and Mizoguchi (1990).</p><p>Jame, Y.-W., Norum, D.I., 1980. Heat and mass transfer in a freezing unsaturated porous medium. Water Resources Research 16, 811–819. https://doi.org/10.1029/WR016i004p00811</p><p>Mizoguchi, M., 1990. Water, heat and salt transport in freezing soil. sensible and latent heat flow in a partially frozen unsaturated soil. University of Tokyo.</p><p> </p>

On One Controllability of the Schrödinger Equation as Coupled with the Atomic-Level Mannesmann Effect

In this paper we outline a certain way of understanding of macroscopically uncontrollable emergence of the so called Mannesmann effect by means of its induced controllable quantum-mechanical background. In other words, we factually present a modus operandi of how to avoid macroscopic models of specific atomic-level cavity origin based consequently on a classical fracture mechanics theory. Under such circumstances, the target solution of the controllable microscopic model cannot be determined, since it can obviously arise only as a macroscopic state of the structurally disturbed rolled metal semi-product during the Mannesmann process. We obtain this irrelevance of the target solution, using a very special kind of control of the famous Schrödinger equation employed as a fundamental model equation here. We show contextually that such control follows from some very elementary aspects of the group theory conditioning a physical meaning of the Schrödinger equation written in a controllable form. We specially emerge primary cyclic groups of symmetry of special solutions to the Schrödinger equation. Their imaginary part is given by a control satisfying the Klein-Gordon equation which can be driven (through a specific avoidance of the cyclic group Z4) into a connection with the characteristic series of primary cyclic groups and/or torsion groups respectively. We obtain a physically controllable special results representing a strange correspondence between a certain LET (Linear Energy Transfer) and “quantum-like” tunnelling interpreted for some “everyday” objects, particularly for the considered Mannesmann piercing process with a torsion known from metallurgy. The process violations are shown and further reflected via a standard finite element method (FEM) simulation.

A Neural Element Method

Methods of artificial neural networks (ANNs) have been applied to solve various science and engineering problems. TrumpetNets and TubeNets were recently proposed by the author for creating two-way deepnets using the standard finite element method (FEM) and smoothed FEM (S-FEM) as trainers. The significance of these specially configured ANNs is that the solutions to inverse problems have been, for the first time, analytically derived in explicit formulae. This paper presents a novel neural element method (NEM) with a focus on mechanics problems. The key idea is to use artificial neurons to form elemental units called neural-pulse-units (NPUs), using which the shape functions can then be constructed, and used in the standard weak and weakened-weak (W2) formulations to establish discrete stiffness matrices, similar to the standard FEM and S-FEM. Theory, formulation and codes in Python are presented in detail. Numerical examples are then used to demonstrate this novel NEM. For the first time, we have made a clear connection in theory, formulations and coding, between ANN methods and physical-law-based computational methods. We believe that this novel NEM fundamentally changes the way of approaching mechanics problems, and opens a window of opportunity for a range of applications. It offers a new direction of research on unconventional computational methods. It may also have an impact on how the well-established weak and W2 formulations can be introduced to machine learning processes, for example, creating well-behaved loss functions with preferable convexity.

ON EVALUATION OF THE THREE-DIMENSIONAL ISOGEOMETRIC BEAM ELEMENT

The exact description of the arbitrarily curved geometries, including conic sections, is an undeniable advantage of isogeometric analysis (IGA) over standard finite element method (FEM). With B-spline/NURBS approximation functions used for both geometry and unknown approximations, IGA is able to exactly describe beams of various shapes and thus eliminate the geometry approximation errors. Moreover, naturally higher continuity than standard C0 can be provided along the entire computational domain. This paper evaluates the performance of the nonlinear spatial Bernoulli beam adapted from formulation of Bauer et al. [1]. The element formulation is presented and the comparison with standard FEM straight beam element and fully three-dimensional analysis is provided. Although the element is capable of geometrically nonlinear analysis, only geometrically linear cases are evaluated for the purposes of this study.

Finite Element Convergence Analysis of a Schwarz Alternating Method for Nonlinear Elliptic PDEs

In this paper, we prove uniform convergence of the standard finite element method for a Schwarz alternating procedure for nonlinear elliptic partial differential equations in the context of linear subdomain problems and nonmatching grids. The method stands on the combination of the convergence of linear Schwarz sequences with standard finite element L-error estimate for linear problems.

Analysis of Bending Ability of Soft Pneu-nets Actuators for Soft Robotics

Background: There has been an increasing interest in the soft pneumatic networks (also referred to as pneu-nets) actuators for soft robotics due to their innately softness, ease of fabrication and high customizability. More and more structures of the soft pneu-nets actuators are reported in various relevant patents and papers. Bending ability of soft pneu-nets actuator is one of key characterizing performance. It is characterized as a function of input air pressure as well as geometrical and material parameters, and influenced by the air pressures and design angle. Objective: In this paper, a new structure soft pneu-nets actuators (with different chambers morphology) was developed. The goal of this paper is to analyze the influences of the air pressures and design angle on the bending ability of the soft pneu-nets actuators with new structure. Method: Firstly, a new structure of soft pneu-nets (adjusting chamber’s shape), based on the soft pneu-nets architecture described previously was developed. Then, the soft pneu-nets actuators were treated with a standard Finite Element Method (FEM) using the Abaqus 6.14 Computer Aided Engineering (CAE) package. Several soft pneu-nets actuators with various design angle were analyzed to investigate the influence of the design angle on the bending ability. In order to investigate the effect of these parameters, the relationship between the angle of bending and these parameters were conducted.Thirdly, the influence of chambers morphology on bending ability of soft pneu-nets actuators could be assessed. Results: When the air pressure P is under 13.5 kPa, the differences of angle of bending under same design angle are not evident, but when the air pressure P is over 13.5 kPa, the differences of angle of bending at same design angle increase; At the same air pressure P, when the air pressure P is under 13.5 kPa, the difference of the angle of the bending between different design angle is less, The effect of gravity is greater than that of the air pressures P. When the air pressure P is over 13.5 kPa, however, the design angle shows more influential on the angle of the bending. Conclusion: The angle of bending increases with the increase of the air pressures P; the chambers with a bigger design angle (thinner inside walls) enabled greater bending at given air pressures P.

Finite element method with local damage of the mesh

We consider the finite element method on locally damaged meshes allowing for some distorted cells which are isolated from one another. In the case of the Poisson equation and piecewise linear Lagrange finite elements, we show that the usual a priori error estimates remain valid on such meshes. We also propose an alternative finite element scheme which is optimally convergent and, moreover, well conditioned, i.e. the conditioning number of the associated finite element matrix is of the same order as that of a standard finite element method on a regular mesh of comparable size.

The use of proper orthogonal decomposition for the simulation of highly nonlinear hygrothermal performance

In this paper, the use of proper orthogonal decomposition for simulating nonlinear heat, air and moisture transfer is investigated via two applications: HAMSTAD benchmarks 2 and 3. Moreover, the potential of the reduced models constructed by proper orthogonal decomposition for simulating new problems with longer simulation periods is assessed. To illustrate the feasibilities of proper orthogonal decomposition method in the field of building physics, the accuracies of the reduced models are compared with the standard finite element method. The outcomes show that with a sufficient number of construction modes and a relatively large amount of snapshots, proper orthogonal decomposition method can deliver accurate results. In addition, guidelines on selecting an appropriate amount of simulation snapshot and construction modes are provided.