Adaptive Mesh Refinement Strategy Research Articles

The adaptive hygrothermo–magneto–electro–elastic coupling improved enriched finite element method for functionally graded magneto–electro–elastic structures

In this study, the hygrothermo–magneto–electro–elastic coupling improved enriched finite element method (HC-IEFEM) is proposed to analyze functionally graded magneto–electro–elastic (FG-MEE) structures. The static behavior of FG-MEE structures in the hygrothermal environment is studied. In the enriched finite element method (EFEM) enhanced by interpolation cover functions, the improved shape functions are added to solve the rank defect problem. Quadrilateral elements are used in the analysis of FG-MEE structures in this study. This improved method has proven to be relatively more efficient and resistant to mesh distortion than the traditional finite element method (FEM). Besides, the adaptive mesh refinement (AMR) scheme is used to refine the mesh in local areas to improve numerical calculation efficiency. Numerical examples show that the adaptive HC-IEFEM can achieve relatively high accuracy for analyzing FG-MEE structures. The proposed HC-IEFEM can obtain high accuracy in the hygrothermal environment using a relatively coarse mesh through the improved shape function and AMR scheme. Therefore, the significant potential is demonstrated in analyzing FG-MEE structures in the hygrothermal environment using the proposed adaptive HC-IEFEM.

A study on the hydroacoustic characterisation of a cavitating propeller by dynamic adaptive mesh refinement technique

Underwater radiated noise (URN) of a marine propeller has received significant interest in recent decades due to its implications on marine fauna. Therefore, an accurate prediction of URN at an early stage of the propeller design is becoming imperative. This study presents a numerical investigation into the noise prediction of a marine propeller, including cavitation and a comparison with experimental test results obtained from the URN database from the King's College D (KCD) standard propeller series. Amongst the propellers tested in the series, the member KCD-193 was chosen to scrutinise in this study due to the significant variance of the cavitation types experienced by this propeller member and consequent characteristic variations observed in its URN spectral response. Numerical URN predictions of different flow conditions, represented by the advance coefficient and cavitation number, were conducted to investigate their effects on the noise spectrum. These predictions were compared with the experimental results to enable interpretation of the impact of various aspects of the simulation on URN prediction accuracy. In this investigation, one of the most prominent noise sources, tip-vortex cavitation (TVC), was identified as a critical aspect that needs to be captured by the numerical simulations for accurate URN predictions using CFD simulations. The influence of TVC on the spectrum was observed to be significant. The inception and stable presence of TVC dominated the frequency response of the broadband hump. In order to address this, a systematic adaptive mesh refinement strategy was implemented based on the vortex criterion to solve the flow characteristics in the propeller slipstream accurately. To further complement this task, a correlation between the cavitation bubble growth and collapse phenomenon by the sensitivity of the broadband hump on the spectrum was established based on the experimental results. The central frequency of the broadband hump was observed to vary with the advance coefficient and cavitation number. The reduction in the cavitation number resulted in a shift of this hump towards lower frequencies. The URN level of the hump decreased slightly in the high frequency by the reduction in the advance coefficient and the developing cavitation, demonstrating the cushioning effect on the spectrum. An accurate assessment of the noise spectrum, as far as numerical predictions are concerned, particularly on the broadband hump frequency bandwidth, was directly associated with the resolution of the TVC.

A Posteriori Error Estimation for Parabolic Problems with Dynamic Boundary Conditions

This paper is concerned with adaptive mesh refinement strategies for the spatial discretization of parabolic problems with dynamic boundary conditions. This includes the characterization of inf–sup stable discretization schemes for a stationary model problem as a preliminary step. Based on an alternative formulation of the system as a partial differential–algebraic equation, we introduce a posteriori error estimators which allow local refinements as well as a special treatment of the boundary. We prove reliability and efficiency of the estimators and illustrate their performance in several numerical experiments.

Low-cycle fatigue crack growth in brittle materials: Adaptive phase-field modeling with variable-node elements

This work presents an adaptive phase field framework for the simulation of crack nucleation and propagation in brittle materials subject to low-cycle loading. We introduce a fatigue history strain parameter within the phase-field framework to capture the fatigue effect. The resulting coupled differential equations are solved using a staggered iteration scheme. In order to improve the computational efficiency of the phase-field model, an adaptive refinement scheme is introduced. This scheme utilizes an artificial threshold that takes into consideration of both phase-field variables and cumulative fatigue history field variables to determine the elements to be refined. The handling of the hanging nodes resulting from mesh refinement is accomplished through the application of the variable-node element technique, offering a flexible and efficient mesh integration technique. We validate our proposed numerical scheme through five representative numerical examples: single-edge cracked specimen, compact tension specimen, double-edge cracked panel, plate with a hole and plate with multi-cracks and hole. The results demonstrate that the adaptive mesh refinement scheme significantly reduces computational costs without compromising the accuracy of the numerical predictions.

New Adaptive Mesh Refinement Strategy for Entry Guidance via Sequential Convex Programming

This study focuses on the entry guidance problem, which is a highly nonlinear and constrained optimal control problem. The entry guidance problem is formulated as a nonlinear trajectory optimization problem with six-degrees-of-freedom dynamics, path constraints, and boundary constraints. The formulated problem is discretized and solved using sequential convex programming (SCP) with successive linearization. However, during the discretization process, conventional collocation methods encounter a challenge with high-frequency jittering in the control and bank angle profiles near active path constraints. Because the considered problem especially involves tight path constraints, the solution jittering issue becomes severe. Therefore, this study proposes a novel mesh refinement strategy to address this problem. Unlike conventional mesh refinement strategies based on interpolation error, the proposed approach resolves the convergence and solution jittering issue of the SCP framework by linking the direct and indirect methods with the costate estimation under successive linearization. Along with the development, we introduce a novel costate estimation method and provide proof of its validity. We present numerical simulation results of the proposed method and conduct a comparison with conventional approaches. The results obtained demonstrate successful mitigation of the solution jittering issue and a significant improvement in solution accuracy regarding compliance with the necessary conditions of optimality.

Boosting galactic outflows with enhanced resolution

ABSTRACT We study how better resolving the cooling length of galactic outflows affect their energetics. We perform radiative-hydrodynamical galaxy formation simulations of an isolated dwarf galaxy ($M_{\star }=10^{8}\, \mbox{M}_\mathrm{\odot }$) with the ramses-rtz code, accounting for non-equilibrium cooling and chemistry coupled to radiative transfer. Our simulations reach a spatial resolution of $18 \, \mathrm{pc}$ in the interstellar medium (ISM) using a traditional quasi-Lagrangian scheme. We further implement a new adaptive mesh refinement strategy to resolve the local gas cooling length, allowing us to gradually increase the resolution in the stellar-feedback-powered outflows, from $\ge 200 \, \mathrm{pc}$ to $18 \, \mathrm{pc}$. The propagation of outflows into the inner circumgalactic medium is significantly modified by this additional resolution, but the ISM, star formation, and feedback remain by and large the same. With increasing resolution in the diffuse gas, the hot outflowing phase ($T \gt {8} \times 10^{4} \, \mathrm{K}$) systematically reaches overall higher temperatures and stays hotter for longer as it propagates outwards. This leads to two-fold increases in the time-averaged mass and metal outflow loading factors away from the galaxy ($r=5\, \mathrm{kpc}$), a five-fold increase in the average energy loading factor, and a ≈50 per cent increase in the number of sightlines with $N_{\rm{O {\small VI}}} \ge 10^{13}\, \mathrm{cm}^{-2}$. Such a significant boost to the energetics of outflows without new feedback mechanisms or channels strongly motivates future studies quantifying the efficiency with which better-resolved multiphase outflows regulate galactic star formation in a cosmological context.

MHD modelling of coronal streamers and their oscillations

Context. The present work investigates solar coronal dynamics in particular streamer waves. Streamer waves are transverse oscillations of the streamer stalk, often generated by the passage of a coronal mass ejection (CME). Recent observational studies infer that the streamer wave is an eigenmode of the streamer plasma slab and an excellent candidate for coronal seismology. Aims. In the present work, we aim to numerically investigate the theoretical concepts of the physics and properties of streamer waves and to complement the observational statistical analysis of these events. Methods. We used the magnetohydrodynamics (MHD) module of MPI-AMRVAC. An adaptive mesh refinement scheme was employed to achieve high resolution for the streamer structure. All the simulations were computed on the same base grid with the same numerical methods. We considered a dipole magnetic field on the Sun and a uniformly accelerating solar wind. We introduced a θ-velocity perturbation within our computational domain in the plane of a streamer to excite the transverse motion. Results. A numerical model for the streamer wave phenomena was constructed in the framework of 2.5D MHD. We performed a parameter study and identified a sensitivity of the streamer dynamics to the background solar wind speed, the characteristics of the perturbation, and the input parameters for the model, such as temperature and magnetic field. We performed a statistical analysis and compared the obtained modelling results with the database of such events from observations from three different coronagraphs. We observed a narrow range of phase speeds and a correlation between wavelength and period. This is consistent with the observations and supports the idea that the streamer wave is an eigenmode of the streamer plasma slab. The measured phase speed is consistently significantly higher than the speed calculated from the measured period and wavelength. The simple fit, when the difference between these two speeds is exactly the background solar wind speed, only matches a small fraction of the data. The obtained results indicate that further investigation is required into the Doppler shift effect in the MHD theory for coronal seismology.

Multiscale modeling of liquid jet breakup in crossflow using an Eulerian/Lagrangian approach

Liquid atomization is a very complex issue, involving multiple length and time scales over several orders of magnitude. To better understand the atomization characteristics of the main injection in a lean premix prevaporize (LPP) combustor, a volume of fluid (VOF)–particle conversion algorithm Lagrangian particle tracking (LPT) coupled approach was proposed to simultaneously reproduce the primary and secondary breakup processes. A VOF model with an adaptive mesh refinement strategy was used to resolve the liquid disintegration on a large scale. The small liquid structures qualified as droplets were transformed into discrete particles based on particle conversion criteria. Next, these particles were tracked using the LPT method to simulate the secondary breakup process. The proposed coupled method used in the Eulerian/Lagrangian framework was validated against liquid jet in crossflow experimental data. The numerical results achieved good agreement with the experimental data. Finally, the proposed method was used to predict the atomization characteristics of the main injection in an LPP combustor under various aerodynamic conditions. Qualitative and quantitative information about liquid deformation and spray characteristics were obtained, which varied depending on the aerodynamic parameters.

Adaptive Mesh Refinement Strategies for Cost-Effective Eddy-Resolving Transient Simulations of Spray Dryers

The use of adaptive meshing strategies to perform cost-effective transient simulations of spray drying processes is evaluated. These simulations are often computationally expensive, given the large differences between the characteristic times of the central jet and those of the unsteady flow developed at its periphery. Managing the computational cost through the control of the grid resolution by regions is inadequate in many of these applications since the grid resolution requirements change dynamically within the domain. These conditions are related to the unsteady nature of the flow in both the central jet and the flow recirculation zones. Therefore, the application of adaptive mesh refinement (AMR) strategies is recommended. In this paper, general AMR criteria based on relative errors are evaluated by testing three mesh adaptation criteria: velocity gradient, pressure gradient, and vorticity. This evaluation is performed using a low-cost turbulence model with eddy resolution (DDES) in two different types of drying chambers, in which experimental measurements are available. The use of AMR exerts appreciable effects on decreasing computational costs and contributes to the capture of large eddies in critical regions. The present approach provides an appropriate balance between solution accuracy and computational cost. By using a correct AMR configuration, it is possible to obtain results similar to those obtained on a fixed grid but reducing the computational costs by 3 to 5 times.

Mesh coarsening using the phantom-node method in the phase field model

The phantom-node method is applied in the phase field model for mesh coarsening to improve computational efficiency. When the adaptive mesh refinement scheme is used, the fine mesh still remains along the path where crack already passed. However, structural properties such as stiffness, damage and strain energy of an element completely partitioned by the crack path remain almost unchanged during the rest of the simulation. Using the phantom-node method, the fine mesh in the crack path domain is recovered into a coarse mesh. This numerical scheme significantly reduces the number of degrees of freedom involved in the computation. Using the mesh coarsening method together with the adaptive mesh refinement and adaptive update schemes, the computational cost is further reduced. Through several numerical examples, the performance of the proposed method is demonstrated.

CAFE-AMR: a computational MHD solar physics simulation tool that uses AMR

ABSTRACT The study of our Sun holds significant importance in space weather research, encompassing a diverse range of phenomena characterized by distinct temporal and spatial scales. To address these complexities, we developed CAFE-AMR, an implementation of an adaptive mesh refinement (AMR) strategy coupled with a magnetohydrodynamics (MHD) equation solver, aiming to tackle solar-physics-related problems. CAFE-AMR employs standard fluid dynamics methods, including finite-volume discretization, HLL and Roe class flux formulas, linear order reconstructors, second-order Runge–Kutta, and corner transport upwind time stepping. In this paper, we present the core structure of CAFE-AMR, discuss and evaluate mesh refinement criteria strategies, and conduct various tests, including simulations of idealized solar wind models, relevant for space weather applications.

Deep reinforcement learning for adaptive mesh refinement

Finite element discretizations of problems in computational physics often rely on adaptive mesh refinement (AMR) to preferentially resolve regions containing important features during simulation. However, these spatial refinement strategies are often heuristic and rely on domain-specific knowledge or trial-and-error. We treat the process of adaptive mesh refinement as a local, sequential decision-making problem under incomplete information, formulating AMR as a partially observable Markov decision process. Using a deep reinforcement learning (RL) approach, we train policy networks for AMR strategy directly from numerical simulation. The training process does not require an exact solution or a high-fidelity ground truth to the partial differential equation (PDE) at hand, nor does it require a pre-computed training dataset. The local nature of our deep RL (DRL) allows the policy network to be trained inexpensively on much smaller problems than those on which they are deployed. The new DRL-AMR method is not specific to any particular PDE, problem dimension, or numerical discretization. The RL policy networks, trained on simple examples, can generalize to more complex problems, and can flexibly incorporate diverse problem physics. To that end, we apply the method to a range of PDEs, using a variety of high-order discontinuous Galerkin and hybridizable discontinuous Galerkin finite element discretizations. We show that the resultant DRL policies are competitive with common AMR heuristics and strike a favorable balance between accuracy and cost such that they often lead to a higher accuracy per problem degree of freedom, and are effective across a wide class of PDEs and problems.

Multi-scale Multi-physics Phase-field coupled Thermo-mechanical approach for modeling of powder bed fusion process

The present study investigates the thermomechanical simulation of powder bed fusion (PBF) process, focusing on the characterization of the melt pool. With the aim of comprehensively understanding the complex multi-physics phenomena inherent to the PBF process, our objective is to establish a robust and refined computational framework that is capable of systematic evaluation of the dynamic evolution of the melt pool geometry and temporal characteristics across the various stages of its formation. For this purpose, the Allen-Cahn phase field formulation is integrated with the elastoplastic material model based on J2 plasticity theory and the model implementation is conducted based on a finite element method (FEM) framework using the Multi-physics Object-Oriented Simulation Environment (MOOSE). As an objective, the model tends to depict the intricate, nonlinear, and non-equilibrium nature of the melt pool, encompassing multi-physics process of the PBF, while assuming powder particles as a homogenized continuum medium and neglecting complex interactions between the particles and their surroundings. This includes the separation of geometry into two domains (i) the powder layer and (ii) the substrate, the melting and solidification of the material, the phase-changing process, and the resulting induced distortion and residual stresses. Additionally, the model incorporates a phase indicator to account for the degree of solidification or consolidation of powder, whose evolution describes diffusion, solidification and shrinkage phenomena. This approach enables the tracking of the movement of the melting front of the metallic material induced by a heat source, effectively dividing the domain into regions of soft and hard solid states with a diffusive interface in between. Furthermore, to enhance the model's fidelity, an adaptive mesh refinement (AMR) strategy is employed to precisely capture important aspects of the phase changing process, such as characteristics of the solidification front and powder melting. To assess the model's performance, it is tested by simulating the processing of a three-dimensional Inconel 625 thin plate, as well as a benchmark example, where the influence of parameters such as temperature and phase change on the mechanical response of the system during the solidification process is elucidated.

Adaptive density-based robust topology optimization under uncertain loads using parallel computing

This work presents an efficient parallel implementation of density-based robust topology optimization (RTO) using adaptive mesh refinement (AMR) schemes permitting us to address the problem with modest computational resources. We use sparse grid stochastic collocation methods (SCMs) for transforming the RTO problem into a weighted multiple-loading deterministic problem at the collocation points. The calculation of these deterministic problems and the functional sensitivity is computationally expensive. We combine distributed-memory parallel computing and AMR techniques to address the problem efficiently. The former allows us to exploit the computational resources available, whereas the latter permits us to increase performance significantly. We propose the parallel incremental calculation of the deterministic problems and the contribution to the functional sensitivity maintaining a similar memory allocation to the one used in the deterministic counterpart. The cumulative computing uses buffers to adapt the evaluation at the collocation points to the parallel computing resources permitting the exploitation of the embarrassing parallelism of SCMs. We evaluate the deterministic problems in a coarse mesh generated for each topology optimization iteration to increase the performance. We perform the regularization and design variable update in a fine mesh to obtain an equivalent design to the one generated in such a mesh. We evaluate the proposal in two- and three-dimensional problems to test its feasibility and scalability. We also check the performance improvement using computational buffers in parallel computing nodes. Finally, we compare the proposal to the same approach using different preconditioners without AMR schemes showing significant performance improvements.

A continuous field adaptive mesh refinement algorithm for isogeometric topology optimization using PHT-Splines

This study proposes a continuous density-field based isogeometric topology optimization (ITO) using Polynomial splines over Hierarchical T-meshes (PHT-Splines). Taking the benefit of isogeometric methods, the design density values for the mesh are defined using spline basis functions as a continuous density function (CDF) through the control points. The material density at the control points will then appear as a facade over the domain, which is further employed to generate a very smooth topology and also to propose a marking criterion for an adaptive mesh refinement (AMR) strategy during the optimization process. During this adaptive meshing, the isogeometric domain mesh is refined during each adaptive step, properly tracking the density variation over the elements and sharpening the boundaries of the optimized topologies. By utilizing the rationale of the Geometry Independent Field approximaTion (GIFT) framework, any complex multi-patch geometry constructed in an industry-standard CAD package using Non-Uniform Rational B-Splines (NURBS) may be imported and discretized using PHT-Splines, including higher dimensions. This helps to maintain the geometrical accuracy through an initial coarse mesh and computational accuracy of the optimization iterations through the local refinement feature of the PHT-Splines. An adaptive first-neighbourhood smoothening algorithm based on the Shepard function is also proposed to obtain manufacturable topologies and to circumvent post-processing stages. The proposed method was observed to provide remarkably smooth topologies with an initial coarse mesh, thereby obtaining a computationally efficient domain to be solved. Compared to existing element density based methods, the proposed method demonstrates a substantial minimization in Degree’s-of-Freedom (DoF) requirement to arrive at clear and realistic solutions. Also, when compared to globally refined solutions, a 50%–90% reduction in DoF is achieved using the adaptive refinement strategy.

High-performance GPU computing of phase-field lattice Boltzmann simulations for dendrite growth with natural convection

The effect of natural convection on dendrite morphology is investigated through three-dimensional large-scale phase-field lattice Boltzmann simulations using a block-structured adaptive mesh refinement scheme with the mother-leaf method in a parallel-GPU environment. The simulations confirmed that downward buoyancy enhances the growth of the primary and secondary arms, and upward buoyancy delays the growth of those arms. In addition, the effect of natural convection on the solidification morphologies gradually decreased as the primary arm tip reached the top of the computational domain and finally stopped. Furthermore, in the longer simulation under purely isothermal diffusive conditions, detachment of the secondary arms owing to curvature-driven fragmentation was observed. A large-scale non-isothermal dendrite growth simulation was also conducted, wherein it was observed that the tip growth rate of the primary arm was delayed, and the secondary arm spacing was larger than that in the isothermal condition.

An embedded strategy for large scale incompressible flow simulations in moving domains

In this work we describe a methodology to approximate the incompressible Navier-Stokes equations in time dependent domains. To deal with the motion of the domain, we employ a fixed mesh method that we call fixed-mesh ALE. It consists of writing the equations in a moving ALE reference system but then projecting them onto a fixed background mesh. This implies that the boundaries of the elements do not necessarily coincide with the physical boundaries, and thus there is the possibility of badly cut elements. We use a Nitsche's type formulation to prescribe the boundary conditions and stabilise the bad cuts by introducing a term that penalises the gradient of the unknown orthogonal to the finite element space in a patch that contains the badly cut element. The flow formulation is a stabilised finite element method that allows one to treat convection dominated flows and to use equal velocity–pressure interpolation. Furthermore, this formulation can be shown to behave as an implicit large eddy simulation approach. A key issue is that the sub-grid scales on which the formulation depends are allowed to be time dependent; this fact has proved to be crucial for the robustness of the approach. The calculation of the velocity and the pressure is segregated by using a fractional step scheme designed at the pure algebraic level, and the conditioning of the pressure equation is improved by using an artificial compressibility technique. Finally, an adaptive mesh refinement strategy is described. All the algorithms are implemented in a parallel environment. The strategy described can be applied to complex flow problems, and in particular here we show the simulation of the air flow generated by a train moving inside a tunnel.

An adaptive mesh refinement strategy for 3D phase modeling of brittle fracture

In order to improve the calculation accuracy and efficiency of the phase field method (PFM), an adaptive mesh refinement strategy suitable for 3D problems is proposed. The refined area is selected in terms of both the phase field value and the phase field increment value. The transition area is then chosen based on the transition element determination algorithm. The corresponding areas are refined using three basic and two advanced refinement templates. The new mesh is made up of hexahedrons and has no hanging nodes. The improved backtracking algorithm based on distance judgment is used in 3D adaptive mesh refinement and has proven its capacity to increase calculation efficiency. The accuracy and applicability of PFM to low-quality meshes with 3D adaptive mesh refinement are verified through benchmark tests, and the errors are all within 1%. Compared with the global refinement mesh, the number of elements in the PF model with 3D adaptive mesh refinement is reduced by more than 80%, and the computational efficiency is increased by 16 times. Combined with virtual crack insertion technology, mode I-II-III fracture is successfully simulated. The accurate and efficient calculation of cracked structures is realized. 3D multi-level adaptive mesh refinement is preliminary developed, and compared with single-level adaptive mesh, the computational efficiency is about two times higher, which provides a potential way to solve the efficient simulation of crack propagation in large-volume structures.

Adaptive isogeometric analysis–based phase-field modeling of interfacial fracture in piezoelectric composites

This paper implements an adaptive phase-field model to investigate the interfacial fracture in transversely isotropic piezoelectric composites under different electromechanical loadings using an isogeometric framework based on polynomial splines over hierarchical T-meshes (PHT-splines). The model introduces the interface and crack phase-field parameters to regularize the sharp interface and crack topologies, respectively and the regularized energy terms are incorporated corresponding to crack and interface evading the need for line integration along them. It also modifies the energy functional to scrutinize the interaction between interfacial damage and crack propagation in piezocomposites and thus is capable of capturing the fracture patterns including interfacial debonding, matrix cracking and their interactions under different electric fields. An adaptive mesh refinement scheme using PHT-splines is implemented to efficiently resolve for interface and crack phase-fields and track the complex crack propagation paths without any ad hoc criteria. The efficacy and robustness of the proposed model are demonstrated using a set of examples simulating the electromechanical fracture in piezoelectric composites. Moreover, different fracture mechanisms in composite materials including the crack nucleation, interfacial debonding, matrix cracking, crack propagation and coalescence are found to be precisely captured using the present modeling technique.

Numerical Study on the Water Entry of a Freely Falling Unmanned Aerial-Underwater Vehicle

The unmanned aerial–underwater vehicle (UAUV) is a new type of vehicle that can fly in the air and cruise in water and is expected to cross the free water surface several times to perform continuous uninterrupted observation and sampling. To analyze the hydrodynamic and motion characteristics of the vehicle, the whole water-entry process of a multi-degree-of-freedom UAUV with various velocity and pitch angle was investigated through a Reynolds-averaged Navier–Stokes method. The computational domain was meshed by trimmer cells. The relative movement between the vehicle and fluid domain was simulated using moving reference frame overset mesh to delineate the interaction region around vehicle body. To reduce the computational cost, adaptive mesh refinement and adaptive time-stepping strategy were used to capture the slamming pressure accurately with reasonable computational effort. First, convergence study is considered. Simulations of the vehicle with various initial velocities and different pitch angles were performed. The variable physical properties were analyzed, and detailed results through the time-varying force and velocity were shown. Initial velocity and pitch angle are found to significantly influence hydrodynamic behavior, including the time-varying force, while thickness ratio has a great impact on added mass and pressure. The results show that higher entry velocity results in greater peak vertical force. The transverse hydrodynamic characteristics for oblique water entry of the vehicle with varies pith angle are quite different.