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one-dimensional. Additionally, analysed tissue is exposed to laser irradi ation, and the internal heat sources resulting from laser irradiation based on the Beer law are taken into account. Moreover, the perfusion rate and the effective scattering coefficient are treated as variables dependent on tissue damage. For numerical calculations, the interval version of the Finite Pointset Method has been used. All calculations are performed due to the direct interval arithmetic rules. The paper is concluded by presenting the obtained results
article is to familiarize the reader with the most important concepts, features and mathematical equations used in particular methods. The article is intended mainly for people who want to get acquainted with the current possibilities of numerical modelling in the field of broadly understood fluid mechanics. The material is intended to facilitate the decision on how to implement the planned play research.
linked by the use of classic multiscale homogenization techniques. Since in bone tissue each micro-scale domain is distinct form its neighbour, applying a classic multiscale homogenization technique to a complete bone structure could represent an inadmissible computational cost. Thus, this work proposes a homogenization methodology that is computationally efficient, presenting a reduced computational cost, and is capable to define the homogenized microscale mechanical properties of the trabecular bone highly heterogeneous medium. Methods: The methodology uses the fabric tensor concept in order to define the material principal directions. Then, using an anisotropic phenomenological law for bone tissue correlating the local apparent density with directional elasticity moduli, the anisotropic homogenized material properties of the micro-scale patch are fully defined. To validate the developed methodology, several numerical tests were performed, measuring the sensitivity of the technique to changes in the micro-patch size and preferential orientation. Results: The results show that the developed technique is robust and capable to provide a consistent material homogenization. Additionally, the technique was combined with two discrete numerical techniques: the finite element method and radial point interpolation meshless method. Conclusions: Structural analyses were performed using real trabecular patches, showing that the proposed methodology is capable to accurately predict the micro-scale patch mechanical behavior in a fraction of the time required by classic homogenization techniques.
the conducted investigations, due to dynamic character of the whole phenomena, characterized by extremely high values of strains and strain’s rate, the meshless explicit approach was used (Smoothed Particle Hydrodynamics method implemented in AUTODYN software). This approach minimalized the negative effects of deformation of “classical” Lagrangian mesh. In order to validate a numerical model, the results were compared with the simplified Gurney’s formula, which provides high accuracy of fragment’s velocity for regular shapes of casing. Comparison of the results showed low value of relative discrepancy (lower than 10%) for the cylindrical part of the casing in which detonation was fully developed and resulted in higher values of relative discrepancy of initial velocity for the non – cylindrical region, especially where the detonation was not developed.
uwadze dynamiczny charakter badanego zjawiska, charakteryzującego się dużymi odkształceniami oraz szybkościami odkształceń rozważanych materiałów, symulacje przeprowadzono z wykorzystaniem bezsiatkowej metody SPH, bazującej na jawnym schemacie numerycznym. Obliczenia przeprowadzono z wykorzystaniem środowiska AUTODYN. Wykorzystana metoda wyeliminowała negatywny wpływ deformacji elementów w klasycznym Lagrange’owskim sformułowaniu modelowania ruchu fazy stałej. Walidacja modelu teoretycznego została przeprowadzona w oparciu wyniki uzyskane przy użyciu wzorów Gurney’a dla rozpatrywanego układu. W obszarze „rozwiniętej detonacji”, stwierdzono satysfakcjonującą dla celów inżynierskich rozbieżność pomiędzy wynikami numerycznymi oraz referencyjnymi (na poziomie mniejszym niż 10 %). Większą rozbieżność pomiędzy wynikami uzyskanymi z zastosowaniem obu podejść uzyskano dla obszarów, których geometria charakteryzowała się stożkowym kształtem ładunku wybuchowego oraz w obszarach, w których detonacja nie rozwinęła się w pełni.
apply Smoothed Particle Hydrodynamics, a meshless approach, to simulate the break-up of a liquid cylinder inside the gaseous phase, i.e. the Rayleigh-Plateau instability. Results obtained in 3D show that even a relatively coarse resolution allows one to predict correctly the size of droplets formed in the process. The detailed analysis of the break-up time in 2D setup implies that a certain level of spatial discretisation needs to be reached to determine this moment precisely.
infinitely smooth radial basis functions such as the multi-quadrics and inverse multi-quadrics, the shape parameter must be chosen properly to obtain accurate approximations while avoiding ill-conditioning of the interpolating matrices. The optimum shape parameter can vary depending on the field, such as in locations of sharp gradients or shocks. Typically, the shape parameter is chosen to maintain a high conditioning number for the interpolation matrix, rendering the RBF smooth [1–10]. However, this strategy fails for a problem with a shock or sharp discontinuity. Instead, in such cases the conditioning number must be kept small. The focus of this work is then to demonstrate the use of RBF interpolation in the approximation of sharp gradients or shocks by use of a RBF blending interpolation approach. This RBF blending interpolation approach is used to maintain the optimum shape parameter depending on the field. The approach is able to sense gradients or shocks in the field and adjust the shape parameter accordingly to keep excellent accuracy. Presented in this work, is an explanation of the RBF blending interpolation methodology and testing of the RBF blending interpolation approach by solving the Burger’s equation using the virtual finite difference method.
main emphasis is laid on appropriate assumption of the interpolant for the sought function due to the number of the boundary conditions in analysed problem. This interpolation function enables to obtain the weighting coefficients for derivative approximation in a governing equation. The method is applied to free vibration analysis of arbitrarily shaped membrane and plate.
automation, stability, and accuracy by blending meshless solution strategies based on a variety of shape functions to achieve stable and accurate iteration process that is integrated with a newly developed, highly adaptive model generation employing quaternary triangular surface discretization for the boundary, a binary-subdivision discretization for the interior, and a unique shadow layer discretization for near-boundary regions. Together, these discretization techniques provide directionally independent, automatic refinement of the underlying native problem model to generate accurate adaptive solutions without need for intermediate user intervention. By coupling the model generation with the solution process, MIMS addresses issues posed by ill-constructed geometric input and pathologies often generated from solid models in the course of CAD design.
wyselekcjonowanych węzłów tworzących gwiazdę różnicową wyznacza się odpowiednie wagi. W artykule przeprowadzono testy numeryczne ilustrujące dokładność omawianej metody w zależności od liczby węzłów gwiazdy, współczynnika kształtu funkcji radialnych, a także stopnia nieregularności siatki. We wspomnianych testach zwrócono również uwagę na uwarunkowanie układów równań pojawiających się w sformułowaniu metody.
difference stars are determined. On the base of numerical tests the dependence of the accuracy on the number of star nodes, shape parameter in the radial function and the degree of irregularity of the node distribution is shown. In these tests the attention is also focused on the conditioning of equation sets that arise from the formulation of the method.
the polynomialinterpolation in the time stepping method. The accuracy and convergence of the numerical results as well as the computational e?ciency of various approaches are compared in numerical test example.
the approximation of a function given in a large number of scattered nodes. In order to show the accuracy of the formulas some derivatives of an exampled function are computed. The influence of the number of nodes as well as the shape parameter of the radial function on the accuracy and condition number of the system is investigated.
skuteczne w aproksymacji funkcji zadanych w dużej liczbie nieregularnie rozmieszczonych punktów. Dla zilustrowania dokładności schematów różnicowych obliczono pochodne przykładowej funkcji oraz zbadano wpływ liczby węzłów i współczynnika kształtu funkcji radialnych na dokładność przybliżenia pochodnej i wskaźnik uwarunkowania układu równań.
These processes can reasonably be described in purely mechanical terms, such as displacements, strains and stresses and therefore can be analysed using established methods of continuum mechanics. We advocate the use of fully non-linear theory of continuum mechanics. We discuss in some detail modelling geometry, boundary conditions, loading and material properties. We consider numerical problems such as the use of hexahedral and mixed hexahedral–tetrahedral meshes as well as meshless spatial discretisation schemes. We advocate the use of Total Lagrangian Formulation of both finite element and meshless methods together with explicit time-stepping procedures. We support our recommendations and conclusions with an example of brain shift computation for intraoperative image registration.
complex forging process is completely simulated with the finite element code LS-DYNA including the local inductive heating phase of the workpiece as well as the rapid cooling process under pressure of the final stub shaft inside the forging dies. A new meshless element formulation denoted as element-free Galerkin method (EFG) is successfully applied to the forging process in addition to the simulation with standard finite elements. The process of the inductive heating is modelled in a simplified way. The simulation results are validated by means of experimentally measured data, showing good agreement.
stopniu, takie parametry jak ciśnienie hydrostatyczne, intensywność naprężenia i maksymalne naprężenie główne. Znanych i używanych w praktyce jest wiele kryteriów pękania plastycznego opartych na tych parametrach. W niniejszej pracy te kryteria są porównywane. Spośród najbardziej popularnych wybrano kryteria pękania przynależne do różnych grup, klasyfikowane według podstaw danego kryterium, a więc kryteria oparte na mikrostrukturze materiału, kształcie pustek lub mechanizmie ich wzrostu. Kryteria oparte na mechanizmie kontinuum pękania, w których bierze się pod uwagę sprzężenie między odkształceniem plastycznym i degradacją materiału poprzez analizę różnych możliwości rozwoju zniszczenia dla rozciągającego i ściskającego stanu naprężenia, dają bardziej poprawną lokalizację obszarów, w których następuje inicjacja pęknięcia.
time-dependent problems, the Laplace transform technique is utilized. The analyzed domain is divided into small subdomains of circular shapes. A unit step function is used as the test function in the local weak form. It leads to Local Integral Equations (LIE) involving a domain-integral only in the case of transient dynamic problems. The Moving Least Squares (MLS) method is adopted for approximating the physical quantities in the LIE. Efficient numerical methods are presented to compute the fracture parameters, namely, the stress intensity factors and the T-stress, for a crack in Functionally Graded Materials (FGM). The path-independent integral representations for stress intensity factors and T-stresses in continuously non-homogeneous FGM are presented.
elastodynamiki, w których zastosowano aparat transformacji Laplace'a. Badany obszar podzielono na małe podobszary kołowe. Jako funkcję testową w lokalnej, słabej postaci zastosowano jednostkową funkcję schodkową, co prowadzi do lokalnych równań całkowych (LIE). Metodę ruchomych najmniejszych kwadratów (MLS) zastosowano do przybliżenia wielkości fizycznych w LIE. Przedstawiono efektywne metody numeryczne wyznaczania parametrów pękania, a w szczególności współczynników koncentracji naprężeń oraz naprężeń T dla szczelin w materiałach funkcjonalnie gradientowych (FGM). Przedstawiono niezależne od drogi całkowania reprezentacje tych parametrów w materiałach FGM o kontynualnie zmieniającej się niejednorodności.
punktami dyskretyzacji.
impact. Concluding remarks and optimization hints.
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