Numerical Formulation Research Articles

Numerical Formulation of Relationship Between Optimized Runner Blade Angle and Specific Speed in a Francis Turbine

Numerical Formulation of Relationship Between Optimized Runner Blade Angle and Specific Speed in a Francis Turbine

Adaptation of an Eddy Current Model for Characterizing Subsurface Defects in CFRP Plates Using FEM Analysis Based on Energy Functional

In this work, a known Eddy Current (EC) model is adapted to characterize subsurface defects in carbon fiber-reinforced polymer (CFRP) plates intended for the civil aerospace industry. The considered defects include delaminations, microcracks, porosity, fiber breakage, and the simultaneous presence of these defects. Each defect is modeled as an additive variation in the material’s electrical conductivity tensor, allowing for a detailed mathematical representation of the defect’s influence on the CFRP’s electromagnetic behavior. The additivity of the variations in the conductivity tensor is justified by the assumption that the defects are not visible to the naked eye, implying that the material does not require non-destructive testing. The adapted EC model admits a unique and stable solution by verifying that all analytical steps are satisfied. To reconstruct 2D maps of the magnetic flux density amplitude, a FEM formulation is adopted, based on the energy functional because it ensures a stable and consistent numerical formulation given its coercivity. Moreover, the numerical approach allows precise and reliable numerical solutions, enhancing the capability to detect and quantify defects. The numerical results show that the obtained 2D maps are entirely superimposable on those highlighting the distribution of mechanical stress states known in the literature, offering a clear advantage in terms of detection costs. This approach provides an effective and economical solution for the non-destructive inspection of CFRP, ensuring accurate and timely defect diagnosis for maintaining structural integrity.

Wave-Current Interaction Effects on the OC4 DeepCwind Semi-Submersible Floating Offshore Wind Turbine

In order to investigate the hydrodynamic performances of semi-submersible type floating offshore wind turbines (FOWTs), particularly the effect of body-wave-current interaction, the OC4 FOWT is considered in the presence of co-existing regular wave and uniform current fields. The wind loads are not considered at this stage. The problem is treated in the framework of potential-flow theory in the frequency domain, assuming waves of small steepness, and the solution is obtained by using a perturbation expansion method for the diffraction potential with respect to the normalized current speed. Analytical and numerical formulations have been used to treat the inhomogeneous free-surface boundary condition involved in the hydrodynamic problem formulation for the derivation of the associated perturbation potential. The hydrodynamic loads were obtained after evaluating the pressure field around the multi-body configuration using three different computer codes. The results from the three computer codes compare very well with each other and with the numerical predictions of other investigators. Finally, the mean second-order drift forces are calculated by superposing their zero-current values with the corresponding current-dependent first-order corrections, with the latter being evaluated using a ‘heuristic’ approach.

Advancing Climate Modeling through High-Performance Computing: Towards More Accurate and Efficient Simulations

A crucial branch of science called climate modeling uses mathematical equations and computer simulations to study and forecast the Earth's climate sys- tem. The main elements of climate modeling, such as general circulation models (GCMs), data assimilation methods, and numerical formulations, are outlined in this paper. GCMs, which include grid point and spectral models, are effective instruments for examining the behavior of the climate. Four-Dimensional Data Assimilation (4D-Var) is one example of a data assimilation technique that in- corporates observational data into models to improve their correctness. Numeri cal methods, ocean dynamics, heat transport, radiative transfer, and atmospheric dynamics are all included in numerical formulations. The simulation of different climate processes is possible because to these mathematical representations. Fur thermore, the detection of precipitation patterns within climate modeling is using machine learning techniques like Random Forest more frequently. This paper highlights the importance of high-performance computing (HPC) in climate modeling, boosting efficiency and simulations, in the context of research technique. Advanced data assimilation and validation techniques are also examined, as well as the influence of high-resolution modeling on small-scale climatic processes. On HPC platforms, accessibility to climate modeling is addressed. It is shown how climate modeling crosses physics, mathematics, computer science, and engineering to be interdisciplinary. A comprehensive understanding of the Earth's intricate climate system gains from the integration of all its parts, from atmospheric dynamics to data assimilation. We explore the consequences of these research approaches, their contribution to enhancing climate prediction models, and the influence of various factors on climatic variables in the debate. Climate modeling becomes an essential tool for studying precipitation patterns and climate change, ultimately improving our comprehension of the complex cli- mate system on Earth.

Dynamic analysis of cracked thick composite shells by the Boundary Element Method

This article presents a numerical formulation based on the Boundary Element Method for the transient dynamic analysis of cracked thick symmetrical composite shells. The integral formulation uses the static fundamental solutions for thick orthotropic symmetric plates and the anisotropic plain elasticity fundamental solution. Domain integrals associated to distributed loads, curvature and inertial terms are evaluated employing the Radial Integration Method. The crack was modeled using the sub-region method. The developed formulation is implemented computationally and validated through the analysis of several proposed examples. The obtained results demonstrate the validity and robustness of the developed formulation.

Experimental and numerical based model formulation for estimation of subgrade resilient modulus using the repeated load CBR test considering in situ state of stress

Characterizing subgrade in terms of resilient modulus is a crucial aspect of flexible pavement design. This paper proposes a methodology and predictive model to estimate the resilient modulus with better consideration of subgrade soils’ in situ stress state using a simple Repeated Load CBR (RLCBR) test. RLCBR tests were conducted on eight subgrade soils at three moisture contents. Numerical studies were conducted by simulating the CBR test in the commercial package LS-DYNA® to understand the stress state under plunger loading concerning field conditions. A new model was proposed for the characterization of subgrade soils based on laboratory RLCBR tests and the FEM, considering the stress state experienced by subgrade soils in the field. The proposed model was validated using data from four other soils and showed good agreement. The study model showed a better predictive capacity for the low plastic subgrade soils than previously developed models. Practicing engineers can use the developed model for estimating the subgrade resilient modulus at the recommended stress state for mechanistic pavement design while understanding the soil’s load-deformation behavior.

Analysis of thermally-induced fracture of Callovo-Oxfordian claystone: From lab tests to field scale

Argillaceous rocks are a suitable host rock for the deep geological disposal of exothermic High-Level Radioactive Waste (HLW) and Spent Fuel (SF). Excess pore pressures develop in this type of rocks when subject to increased temperatures that, in some circumstances, may lead to the fracturing of the rock. The paper explores this phenomenon by means of coupled numerical analyses carried out within a fully coupled thermo-hydro-mechanical (THM) framework. The general THM formulation is described as well as the anisotropic elastic and anisotropic elastoviscoplastic constitutive laws employed. Thermal extension triaxial tests are simulated as a check on the performance of the numerical formulation and to provide calibration data for the tensile strength of the material. Selected results from a comprehensive set of three-dimensional analyses of a large-scale field heating test, designed to study the possibility of thermal-induced failure in the rock, are presented and discussed. The analyses reproduce satisfactorily the observed patterns of behaviour. The effects of the constitutive law, material parameters and the presence of the excavation damage zone (EDZ) around the main drift and around the heater boreholes are studied. In particular, their effects on the state of the stress in the heated area are examined in the context of the potential for thermal fracturing of the rock.

Study on fracture of hyperelastic Kirchhoff-Love plates and shells by phase field method

Thin walled structures such as plates and shells are widely used in many engineering fields. To Predict its fracture behavior is of great significance for integrity design and strength evaluation of engineering structures. Numerical simulation of the fracture behavior of hyperelastic plates and shells is a challenge due to complex kinematic description, hyperelastic constitutive relationship, geometric nonlinearity and the degradation on elastic parameter caused by fracture damage. Combining Kirchhoff Love (K-L) shell theory with the fracture phase field method, and numerically discretizing the first and second order partial derivatives of displacement field and phase field by using T-splines and meeting the requirements of K-L plate and shell theory for the C1 continuity of the shape function, a model for the isogeometric analysis numerical formulation of the phase field fracture in hyperelastic K-L plates and shells is established. The fracture failure behavior of hyperelastic K-L plates and shells under the uniform load and displacement load is simulated, and the effect of the Gaussian curvature on the fracture behavior of hyperelastic K-L shells is studied. The simulation results show that the present numerical scheme can effectively capture the complex crack propagation path of plates and shells under the uniform load, and the displacement field can effectively reflect the crack distribution of materials. The thin shell with negative Gaussian curvature shows the excellent fracture performance under the internal pressure, and can withstand the greater internal pressure.

A finite volume formulation for one-dimensional Stefan problems

This paper presents a numerical formulation that combines the finite-volume method with a time-dependent boundary-fitted coordinate system to tackle the moving boundary problems for the one-dimensional (1-D) heat equation. Following the formulation process, an algorithm was created with the help of the linear algebra package Eigen to generate the data for the numerical solution. Four benchmark cases of heat conduction were investigated by this approach and compared to the analytical solution: a fixed boundary case, the liquid phase of a melting process, the solid phase of a melting process, and an ablating process. The results demonstrate the capability of the presented numerical formulation in efficiently solving the general 1-D heat problems, especially the one-phase Stefan problems, as well as the consistency of the numerical and analytical solutions. Future works can strive to extend the formulations to Stefan problems of two phases or higher dimensions. The work can then be extended to real-life applications, such as the spacecraft re-entry ablation process, and the process of iceberg melting, demonstrating its critical value in solving practical problems.

Nonlinear behavior of existing pre-tensioned concrete beams: Experimental study and finite element modeling with the constitutive Serial-Parallel rule of mixtures

The performance of existing prestressed concrete (PC) beams and their time-dependent behavior have been concerns for bridge maintenance services. In this paper, laboratory tests of three PC void slab beams taken away from an obsolete bridge were conducted and an alternative more efficient finite element (FE) method was developed. In preparing the three beam specimens, two retained their original reinforced concrete (RC) overlay and one chiseled off its RC overlay. Through the four-point loading tests, the load response plots of the beams from elastic to plastic failure were carefully documented, including the strains along the mid-span section, the strains of longitudinal rebars and strands, and the mid-span deflection. The test results showed that no interface slip was observed between the RC overlay and the top of void slab during the whole loading process, and the yield load of the test beams with the overlay was approximately 45 % higher than that without the overlay. Regarding the proposed nonlinear numerical formulation, the modelling process avoided the treatment of complex reinforcement and prestressing tendon layouts. The prestressed concrete was simulated as a composite material whose behavior is depicted by a constitutive serial-parallel rule of mixtures (SP-RoM). The generalized Maxwell model and generalized Kelvin model were applied to calculate the time-dependent losses of prestress. All the developments have been implemented within the open-source FE code Kratos-Multiphysics environment and are fully available. Compared with the test outcomes, it shows that the numerical results are in good agreement with the solutions in terms of stiffness, load-deflection curves, and damage evolution. The differences between the experimental and numerical values of ultimate load for the specimens with and without RC overlay were all within 10 %. Furthermore, the proposed nonlinear models can also predict the time-dependent prestress losses of existing PC beams.

A Level-Set/FEM approach for particle bed growth in Stokes–Darcy suspension filtration

This article presents an innovative approach to modeling particle bed growth through suspension dead-end filtration, a process of paramount importance in various industrial and environmental contexts. More precisely, it focuses on the numerical formulation of a continuous approach based on a Stokes–Darcy coupling. In a Level-Set/FEM frame, this model effectively captures the dynamic evolution of particle beds (cake), taking into account the interaction between suspended particles and porous media. Model validation is also a key focus of this paper. Numerical tests are carried out under various conditions, and their results are compared with analytical solutions and experimental data drawn from the literature. This permits corroboration of the predictive capabilities of the model, confirming the robustness and accuracy of the proposed approach. Firstly, Stokes–Darcy coupled flows are investigated in simplified cases such as rectangular geometries and coaxial cylinders. These scenarios serve as benchmarks to assess the model performance under simple and varied conditions. Additionally, the more challenging case of a three-dimensional anisotropic flow between two ellipsoids is addressed, where the evolution of the cake takes place in three dimensions. Through rigorous analysis and comparison with analytical solutions, the model is proved efficient in capturing the inherent complexities of such scenarios. Finally, richer two-dimensional Stokes–Darcy flows are considered, in the presence of both impermeable and permeable obstacles, representing a crucial step towards modeling real industrial processes. This last study highlights not only the formation of particle-free zones but also the practical relevance of this work.

Three-dimensional vibration analysis of multilayered composite and functionally graded piezoelectric plates and shells

In the present paper, a coupled 3D exact electro-elastic shell model for Functionally Graded (FG) and composite piezoelectric structures is proposed. Primary variables of the electro-elastic model are the electric potential and the three displacement components. The model allows to evaluate the piezoelectric effect in terms of frequencies and vibration modes. Both closed and open circuit configurations are analyzed and compared. The 3D equilibrium equations and the 3D divergence electric displacement equation for spherical shells give the set of partial differential equations for the electro-elastic problem. The proposed model for spherical shells automatically degenerates into simpler models for plates and cylindrical shells via properly considerations on radii of curvature along in-plane directions. The orthogonal mixed curvilinear coordinates α, β and z are employed. The partial differential governing equations have constant coefficients considering fictitious layers and they are solved using the Navier harmonic form and the exponential matrix method. These features lead to an exact solution for simply-supported boundary conditions. Free vibration analyses are conducted and circular frequencies for the first three thickness vibration modes are computed. After a global assessment phase to verify the correctness of the developed model, new benchmarks are proposed: different thickness ratios and material configurations are investigated. The present work can be intended as a reference general solution for those scientists interested in the study of piezoelectric structures via 2D analytical and numerical formulations.

Modeling damped spring vibration using python to train students' critical thinking and scientific reasoning

Programming has been carried out to model the vibration of a mass spring system with no damping and with damping variations using Python programming language coding. This research aims to simulate the simple harmonic vibration of a mass spring (without damping) and the vibration of a damped mass spring. The programming is designed after formulating the equations of motion of damped mass spring vibrations that behave as second-order differential equations and analyzing the numerical formulation of the Feynman-Newton algorithm arrangement. The method used is experimentation using python to simulate the vibration of the spring. Simulation by varying the damping constant (c = 0, c < √4mk, c = √4mk, and c > √4mk). The simulation results show various vibration graphs, namely simple harmonic vibration, under damped vibration, critically damped vibration, and over damped vibration. The shape of the vibration graph is influenced by the mass, spring constant, and damping constant. The greater the damping constant, the less the maximum speed of vibration. This research has succeeded in visualizing the simple harmonic vibration of a massless spring (without damping) and the vibration of a damped massless spring. This modeling can help the physics learning process in high school in understanding the concept of spring vibration to obtain critical reasoning in accordance with physical phenomena. Modeling of damped spring vibrations using Python can be used as a physics learning media on vibration material to train scientific and critical reasoning skills, and as a learning innovation because it is a new thing where the curriculum and high school learning in Indonesia are not used to being delivered.

Resolution of multiple semi-infinite delaminations using lock-in infrared thermography

Delaminations are flat subsurface defects parallel to the sample surface. Recently we have demonstrated that lock-in infrared thermography, with optical excitation, allows sizing the geometrical parameters (length, depth and thickness) of a semi-infinite delamination. Here, we analyse the ability of this technique to resolve several parallel and semi-infinite delaminations. First, we develop an analytical method (based on the thermal quadrupoles) together with a numerical formulation to calculate the surface temperature of a sample containing several semi-infinite parallel delaminations. We verify that both methods provide the same temperature values, indicating their consistency. Then, we study the ability of lock-in infrared thermography to resolve two close delaminations. In particular we focus on two main configurations: two non-overshadowed delaminations and two superimposed delaminations. Next, after analysing the inverse problem in terms of residual function minimization, we develop a dedicated parametric estimation procedure able to retrieve the geometry of the studied defects. Finally, we test this procedure with synthetic temperature amplitude and phase data to retrieve the geometrical parameters of both delaminations.

Finite strain modelling of Shape Memory Alloys in the logarithmic strain space: A comparative study with other finite strain approaches

Finite strain models for Shape Memory Alloys (SMAs) offer significant advantages over small strain models in that they excel in their ability to characterize SMA behaviour under large strain and finite rotations while eliminating the inherent assumptions associated with the small strain framework. The paper at hand presents a numerical model for SMAs at finite strain, employing the logarithmic strain measure to obtain an additive split of the total strain into its elastic and inelastic (transformation) parts. Notably, the model maintains the thermodynamic consistency validated using the Clausius–Duhem inequality. Additionally, we compare this model with two other finite strain SMA models employing logarithmic strain measures: a hypoelastic model, which splits the rate of deformation tensor additively, and a hyperelastic model, which decomposes the deformation gradient multiplicatively. The numerical formulation for the three finite strain models is specifically tailored to address superelasticity in SMAs. It is derived from the established small strain model and seamlessly extended into the domain of finite strain framework. Several numerical examples are carried out to showcase the logarithmic strain space approach applied to SMAs and to compare its response with that of the other aforementioned finite strain models in SMAs. The meticulous examination reveals that the logarithmic strain space approach aligns well with the existing finite strain models, demonstrating its effectiveness and compatibility.

Wave setup estimation at regional scale: Empirical and modeling-based multi-approach analysis in the Mediterranean Sea

Wave setup is a physical process that induces a temporal increase of the mean water level due to wave dissipation by bottom friction and breaking in the surf zone, extending over tens to hundreds of meters in the cross-shore direction. Wave setup contribution to coastal sea level solely induced by wind and atmospheric effects can increase by more than 100% under extreme events and conditions favoring its formation. It is therefore crucial to consider this phenomenon when assessing sea-level-related coastal hazards. Previous studies estimated the wave setup effect by means of numerical modeling and empirical formulations at regional and global scale. Such analyses require either high computational capacity to implement high-resolution numerical models over large domains, and/or accurate information on coastal morphological features from global or regional databases. Although the Mediterranean Sea is a fetch-limited environment, waves generated from extra-tropical cyclones are powerful enough for wave setup to develop, and subsequently for a potential significant wave setup contribution to extreme coastal sea level. Through the use of both numerical and empirical methods, we investigate the uncertainty associated to wave setup representation on the frequency and magnitude of coastal extreme sea levels occurring on sandy beaches in the Mediterranean Sea. Wave setup values are compared at beach scale between process-based modeling and empirical approaches, showing highly variable results. We also quantify the impact of wave setup on return levels of coastal sea level extremes using reconstructed sea levels. We employ various methods to calculate the wave setup component. Results show high spatial dispersion, with clear differences between the numerical and empirical approaches, especially in regions prone to the development of energetic waves. The total inter-method dispersion of 100-year return levels is often higher than 30 cm for average values of 62.4 cm. We emphasize the important limitations related to wave setup modeling (i.e., its underestimation) at large scale, and call for caution when applying empirical formulations (generally developed from local studies) at regional to global scale, which can lead to unrealistic wave setup values.

Fourth-order exponential time differencing Runge–Kutta scheme and local meshless method to investigate unsteady diffusion–convection problems of anisotropic functionally graded materials

The current article is devoted to proposing a local meshfree method for simulating the unsteady diffusion–convection problems of anisotropic functionally graded materials (AFGM). The first stage of the proposed numerical algorithm is based on discretizing the spatial variables by utilizing the direct meshless local Petrov–Galerkin (DMLPG) method. Then a system of ODEs is extracted which is associated to the time variable. Furthermore, the fourth-order exponential time differencing Runge–Kutta method is applied to solve the system of ODEs. The main aim of this work is to introduce a flexible and low-cost numerical procedure to solve unsteady diffusion–convection problems of AFGM on the complicated geometry. The new numerical technique has admissible resultants to simulate the studied model. This numerical formulation is applied to several examples with complex domains to check its ability.

Unified approach to artificial compressibility and local low Mach number preconditioning

This paper presents a unified approach to artificial compressibility and local low Mach number preconditioning methods. The numerical formulation is presented, which provides a seamless transition between incompressible and compressible systems. A single preconditioner that can act as the artificial compressibility or the local preconditioning under a given Mach number condition is derived. An artificial compressibility method is proposed for the unified system. The unified preconditioner is applied to the upwind approach, Roe's approximate Riemann solver, and the JST artificial dissipation approach. A Roe average, which can be applied to the unified systems in a single form is derived. The preconditioning matrix is applied to a single solver to analyze problems ranging from incompressible fluid flows to compressible flows with shocks.

A computationally efficient approach to model reactive transport during CO2 storage in naturally fractured saline aquifers

Energy storage is critical to any decarbonization, energy sustainability, and energy security strategy. Natural gas, CO2, and hydrogen will compete for sizeable geological storage sites in the future, and society needs to consider all possible storage sites. Saline aquifers and depleted hydrocarbon reservoirs have emerged as key sites for storage. While carbonate rocks possess significant storage potential, as evidenced by successful CO2-enhanced oil recovery (CO2-EOR) projects, most operational and planned Carbon Capture and Storage (CCS) initiatives primarily focus on sandstones. Consequently, extensive fundamental and applied research is necessary to advance technologies for carbonate rock utilization and capitalize on the storage capacities of vast saline aquifers and infrastructure available in near-abandoned depleted carbonate reservoirs. However, the challenges posed by the fracture systems and the substantial heterogeneity inherent to such reservoirs necessitate a comprehensive understanding of rock integrity and the role of fracture networks in the overall storage process. Moreover, modeling CCS in carbonates introduces additional complexities due to reactive minerals, such as calcite and dolomite, which can undergo reactions with injected CO2. Consequently, a numerical formulation capable of producing accurate results within a reasonable computational time is required, considering computationally intensive processes like fracture flow and geochemical reactions. In this work, three approaches to modeling naturally fractured media were assessed, such as the explicit representation of fractures through local grid refinement (LGR), the traditional dual permeability/dual porosity model (DPDK), and the embedded discrete fracture model (EDFM). The combination of EDFM with dynamic gridding in the matrix gridblocks demonstrated superior computational efficiency compared to LGR and DPDK simulations. This novel approach was 75 times faster than LGR and 37 times faster than DPDK while yielding comparable results in terms of different CO2 retention forms (free supercritical fluid, dissolved in water, and residual trapped gas) in one of the geological models investigated. Therefore, it was proposed a novel combination of EDFM and dynamic gridding as a suitable and accurate numerical approach to simulate reactive CO2 transport in fractured saline aquifers. This approach offers a practical means of efficiently evaluating carbonate reservoir targets for CCS projects, enabling effective decision-making in energy storage endeavors.

A new Hermite finite element for nonlinear Kirchhoff rods: The plane case

AbstractThis article presents the formulation of a finite element method for nonlinear Kirchhoff rods, based on a interpolation of the rod's geometry in terms of Hermite shape functions. The critical use of the same interpolation scheme for both the geometry and the kinematics of the rod is shown to lead to the correct invariant properties of the final numerical formulation, thus leading to the correct resolution of the fundamental equilibrium relations along it (balance of forces and moments). The so‐called “self‐straining” is completely avoided. Several numerical examples are presented illustrating the adequacy of the proposed formulation for the analysis of thin rods undergoing large finite deformations.