Abstract: We introduce the nonconforming Virtual Element Method (VEM) for the approximation of second order elliptic problems. We present the construction of the new element in two and three dimensions, highlighting the main differences with the conforming VEM and the classical nonconforming finite element methods. We provide the error analysis and establish the equivalence with a family of mimetic finite difference methods. Numerical experiments verify the theory and validate the performance of the proposed method.

Abstract: This paper introduces a weak Galerkin (WG) finite element method for the Stokes equations in the primal velocity-pressure formulation. This WG method is equipped with stable finite elements consisting of usual polynomials of degree k?1 for the velocity and polynomials of degree k?1 for the pressure, both are discontinuous. The velocity element is enhanced by polynomials of degree k?1 on the interface of the finite element partition. All the finite element functions are discontinuous for which the usual gradient and divergence operators are implemented as distributions in properly-defined spaces. Optimal-order error estimates are established for the corresponding numerical approximation in various norms. It must be emphasized that the WG finite element method is designed on finite element partitions consisting of arbitrary shape of polygons or polyhedra which are shape regular.

Abstract: We consider initial/boundary value problems for the subdiffusion and diffusion-wave equations involving a Caputo fractional derivative in time. We develop two fully discrete schemes based on the piecewise linear Galerkin finite element method in space and convolution quadrature in time with the generating function given by the backward Euler method/second-order backward difference method, and establish error estimates optimal with respect to the regularity of problem data. These two schemes are first- and second-order accurate in time for both smooth and nonsmooth data. Extensive numerical experiments for two-dimensional problems confirm the convergence analysis and robustness of the schemes with respect to data regularity.

Abstract: In this work a theoretical framework implementing the phase-field approach to fracture is used to couple the physics of flow through porous media and cracks with the mechanics of fracture. The main modeling challenge addressed in this work, which is a challenge for all diffuse crack representations, is on how to allow for the flow of fluid and the action of fluid pressure on the aggregate within the diffuse damage zone of the cracks. The theory is constructed by presenting the general physical balance laws and conducting a consistent thermodynamic analysis to constrain the constitutive relationships. Constitutive equations that reproduce the desired responses at the various limits of the phase-field parameter are proposed in order to capture Darcy-type flow in the intact porous medium and Stokes-type flow within open cracks. A finite element formulation for the solution of the governing model equations is presented and discussed. Finally, the theoretical and numerical model is shown to compare favorably to several important analytical solutions. More complex and interesting calculations are also presented to illustrate some of the advantageous features of the approach.

Abstract: A three-dimensional finite element model is proposed to study the effects of laser power and scan speed on the thermal behavior and melting/solidification mechanism during selective laser melting (SLM) of TiC/Inconel 718 powder system. The cooling time during powder delivery is taken into account to simulate the actual production process well. It shows obviously the existence of heat accumulation effect in SLM process and, the tailored set of cooling time of 10 ms during powder delivery alleviates that effectively. The maximum temperature gradient in the molten pool slightly increases from 1.30×104 °C/mm to 2.60×104 °C/mm as the laser power is increased from 75 W to 150 W. However, it is negligibly sensitive to the variation of scan speed. There is a positive corresponding relationship between the maximum rate of temperature change and processing parameters. A low laser power (75 W) or a high scan speed (300 mm/s) is more energy efficient in Z-direction of the molten pool, giving rise to a deep-narrow cross section of the pool. Whereas, a high laser power (150 W) or a low scan speed (50 mm/s) causes a shallow-wide cross section of the molten pool, meaning it is more energy efficient in the Y-direction of the melt. The combination of a laser power of 125 W and a scan speed of 100 mm/s contributes to achieve a sound metallurgical bonding between the neighbor layers and tracks, due to the proper molten pool size (width: 109.3 µm; length: 120.7 µm; depth: 67.8 µm). The SLM experiments on TiC/Inconel 718 powder system are performed to verify the reliability and accuracy of the physical model and, simulation results are proved to be correct.

Abstract: A curved beam element based on the Timoshenko model and non-uniform rational B-splines (NURBS) interpolation both for geometry and displacements is presented. Such an element can be used to suitably analyse plane-curved beams and arches. Some numerical results will explore the effectiveness and accuracy of this novel method by comparing its performance with those of some accurate finite elements proposed in the technical literature, and also with analytical solutions: for the cases where such closed-form solutions were not available in the literature, they have been computed by exact integration of the governing differential equations. It is shown that the presented element is almost insensitive to both membrane- and shear-locking, and that such phenomena can be easily controlled by properly choosing the number of elements or the NURBS degree.

Abstract: Thermally induced residual stresses and residual distortions in the additive manufactured (AM) parts are two of the major obstacles that are preventing AM technology from gaining wide adoption. In this work, a three-dimensional thermo-elastic-plastic model is proposed to predict the thermomechanical behavior in the laser engineered net shaping (LENS) process of Ti-6Al-4V using Finite Element Method (FEM). It is shown that the computed thermal history and mechanical deformations are in good agreement with the experimental measurements. The main contributions of this study are: (I) in the past, a point-wise comparison between simulation results and experimental measurements is more favored to validate the employed model, where the general picture is lost; rather, to validate the proposed model, the simulated distortion of the bottom surface of a thin substrate is compared with experimental measurements using a 3D laser scanner, in terms of both magnitude and distribution map. (II) Rather few works have been done to show the effectiveness of widely employed quasi-static mechanical analysis in the transient LENS process; as such, both quasi-static and dynamic simulations are performed and compared mechanically to demonstrate the validity of using quasi-static modeling to save computational cost.

Abstract: In natural environment, many biological structures are tubular and exhibit excellent mechanical properties that can reduce self-weight effectively and transport more water and nutrients, such as bamboo. In this paper, the structure of bamboo was introduced to increase the axial and lateral energy absorption of thin-walled tubes by using bionic design method. Energy absorption ability of bamboo was tested by drop-weight experiments. The results showed that the energy absorption was excellent due to the gradient distribution of vascular bundles, nodes and density. These advantages of the bamboo make it possible to design of bionic structure which composed of 1 bionic node and 3 bionic inner tubes with 18, 9 and 4 bionic elements in each inner tube. Numerical examples of bionic structures under axial/lateral impacts were solved with nonlinear finite element method (FEM). The results indicated that the bionic design enhances the specific energy absorption (SEA) of tubes. Thus, the bionic structure is exactly excellent energy absorption under lateral/axial impact and can be used in the future.

Abstract: The finite element method is used to approximately solve boundary value problems for differential equations. The method discretizes the parameter space and finds an approximate solution by solving a large system of linear equations. Here we investigate the extent to which the finite element method can be accelerated using an efficient quantum algorithm for solving linear equations. We consider the representative general question of approximately computing a linear functional of the solution to a boundary value problem and compare the quantum algorithm's theoretical performance with that of a standard classical algorithm---the conjugate gradient method. Prior work claimed that the quantum algorithm could be exponentially faster but did not determine the overall classical and quantum run times required to achieve a predetermined solution accuracy. Taking this into account, we find that the quantum algorithm can achieve a polynomial speedup, the extent of which grows with the dimension of the partial differential equation. In addition, we give evidence that no improvement of the quantum algorithm can lead to a superpolynomial speedup when the dimension is fixed and the solution satisfies certain smoothness properties.

Abstract: Simulating the physical behaviors of deformable objects and fluids has been an important topic in computer graphics. While the Lagrangian Finite Element Method (FEM) is widely used for elasto-plastic solids, it usually requires additional computational components in the case of large deformation, mesh distortion, fracture, self-collision and coupling between materials. Often, special solvers and strategies need to be developed for a particular problem. Recently, the hybrid Eulerian/Lagrangian Material Point Method (MPM) was introduced to the graphics community. It uses a continuum description of the governing equations and utilizes user-controllable elasto-plastic constitutive models. The hybrid nature of MPM allows using a regular Cartesian grid to automate treatment of self-collision and fracture. Like other particle methods such as Smoothed Particle Hydrodynamics (SPH), topology change is easy due to the lack of explicit connectivity between Lagrangian particles. Furthermore, MPM allows a grid-based implicit integration scheme that has conditioning independent of the number of Lagrangian particles. MPM also provides a unified particle simulation framework similar to Position Based Dynamics (PBD) for easy coupling of different materials. The power of MPM has been demonstrated in a number of recent papers for simulating various materials including elastic objects, snow, lava, sand and viscoelastic fluids. It is also highly integrated into the production framework of Walt Disney Animation Studios and has been used in featured animations including Frozen, Big Hero 6 and Zootopia.

Abstract: The stress-induced martensitic transformation in tensioned nickel-titanium shape-memory alloys proceeds by propagation of macroscopic fronts of localized deformation. We used three-dimensional synchrotron x-ray diffraction to image at micrometer-scale resolution the grain-resolved elastic strains and stresses in austenite around one such front in a prestrained nickel-titanium wire. We found that the local stresses in austenite grains are modified ahead of the nose cone–shaped buried interface where the martensitic transformation begins. Elevated shear stresses at the cone interface explain why the martensitic transformation proceeds in a localized manner. We established the crossover from stresses in individual grains to a continuum macroscopic internal stress field in the wire and rationalized the experimentally observed internal stress field and the topology of the macroscopic front by means of finite element simulations of the localized deformation.

Abstract: The phase field method is a versatile simulation framework for studying initiation and propagation of complex crack networks without dependence to the finite element mesh. In this paper, we discuss the influence of parameters in the method and provide experimental validations of crack initiation and propagation in plaster specimens. More specifically, we show by theoretical and experimental analyses that the regularization length should be interpreted as a material parameter, and identified experimentally as it. Qualitative and quantitative comparisons between numerical predictions and experimental data are provided. We show that the phase field method can predict accurately crack initiation and propagation in plaster specimens in compression with respect to experiments, when the material parameters, including the characteristic length are identified by other simple experimental tests.

Abstract: We present a novel framework for Finite Element Particle-in-Cell methods based on the discretization of the underlying Hamiltonian structure of the Vlasov-Maxwell system. We derive a semi-discrete Poisson bracket, which retains the defining properties of a bracket, anti-symmetry and the Jacobi identity, as well as conservation of its Casimir invariants, implying that the semi-discrete system is still a Hamiltonian system. In order to obtain a fully discrete Poisson integrator, the semi-discrete bracket is used in conjunction with Hamiltonian splitting methods for integration in time. Techniques from Finite Element Exterior Calculus ensure conservation of the divergence of the magnetic field and Gauss' law as well as stability of the field solver. The resulting methods are gauge invariant, feature exact charge conservation and show excellent long-time energy and momentum behaviour. Due to the generality of our framework, these conservation properties are guaranteed independently of a particular choice of the Finite Element basis, as long as the corresponding Finite Element spaces satisfy certain compatibility conditions.

Abstract: To understand the thermal insulation and failure problems of the thermal barrier coatings (TBCs) deeply is vital to evaluate the reliability and durability of the TBCs. Actually, experimental methods can not reflect the real case of the TBCs during its fabrication and service process. Finite element modeling (FEM) play an important role in studying these problems. Especially, FEM is very effective in calculating the thermal insulation and the fracture failure problems of the TBCs. In this paper, the research progress of the FEM on the study of the thermal insulation and associated failure problems of the TBCs has been reviewed. Firstly, from the aspect of the investigation of the heat insulation of the TBCs, the thermal analysis via FEM is widely used. The effective thermal conductivity, insulation temperature at different temperatures of the coating surface considering the thermal conduct, convection between the coating and the environment, heat radiation at high temperature and interfacial thermal resistance effect can be calculated by FEM. Secondly, the residual stress which is induced in the process of plasma spraying or caused by the thermal expansion coefficient mismatch between the coating and substrate and the temperature gradient variation under the actual service conditions can be also calculated via FEM. The solution method is based on the thermal–mechanical coupled technique. Thirdly, the failure problems of the TBCs under the actual service conditions can be calculated or simulated via FEM. The basic thought is using the fracture mechanic method. Previous investigation focused on the location of the maximum residual stress and try to find the possible failure positions of the TBCs, and to predict the possible failure modes of the TBCs. It belonged to static analysis. With the development of the FEM techniques, the virtual crack closure technique (VCCT), extended finite element method (XFEM) and cohesive zone model (CZM) have been used to simulate the crack propagation behavior of the TBCs. The failure patterns of the TBCs can be monitored timely and dynamically using these methods and the life prediction of the TBCs under the actual service conditions is expected to be realized eventually.

Abstract: The conventional $RC$ -lumped thermal networks are widely used to estimate the temperature of power devices, but they lack of accuracy in addressing detailed thermal behaviors/couplings in different locations and layers of the high-power insulated gate bipolar transistor (IGBT) modules. On the other hand, a finite-element (FE)-based simulation is the other method, which is often used to analyze the steady-state thermal distribution of IGBT modules, but it is not possible to be used for a long-term analysis of load profiles of power converter, which is needed for reliability assessments and better thermal design. This paper proposes a novel 3-D $RC$ -lumped thermal network for the high-power IGBT modules. The thermal coupling effects among the chips and among the critical layers are modeled, and boundary conditions, including the cooling conditions, are also considered. It is demonstrated that the proposed thermal model enables both accurate and fast temperature estimation of high-power IGBT modules in the real loading conditions of the converter while maintaining the critical details of the thermal dynamics and thermal distribution. The proposed thermal model is verified by both the FE-based simulation and the experimental results.