In the optimal control problem of nonlinear dynamical system,the Hamiltonian formulation is useful and powerful to solve an optimal control force.However,the resulting Euler-Lagrange equations are not easy to solve,wh...In the optimal control problem of nonlinear dynamical system,the Hamiltonian formulation is useful and powerful to solve an optimal control force.However,the resulting Euler-Lagrange equations are not easy to solve,when the performance index is complicated,because one may encounter a two-point boundary value problem of nonlinear differential algebraic equations.To be a numerical method,it is hard to exactly preserve all the specified conditions,which might deteriorate the accuracy of numerical solution.With this in mind,we develop a novel algorithm to find the solution of the optimal control problem of nonlinear Duffing oscillator,which can exactly satisfy all the required conditions for the minimality of the performance index.A new idea of shape functions method(SFM)is introduced,from which we can transform the optimal control problems to the initial value problems for the new variables,whose initial values are given arbitrarily,and meanwhile the terminal values are determined iteratively.Numerical examples confirm the high-performance of the iterative algorithms based on the SFM,which are convergence fast,and also provide very accurate solutions.The new algorithm is robust,even large noise is imposed on the input data.展开更多
Many mechanical problems can be induced from differential equations with boundary conditions; there exist analytic and numerical methods for solving the differential equations. Usually it is not so easy to obtain anal...Many mechanical problems can be induced from differential equations with boundary conditions; there exist analytic and numerical methods for solving the differential equations. Usually it is not so easy to obtain analytic solutions. So it is necessary to give numerical solutions. The reproducing kernel particle （RKP） method is based on the Carlerkin Meshless method. According to the Sobolev space and Fourier transform, the RKP shape function is mathematically proved in this paper.展开更多
Using the method of undetermined coefficients, we construct a set of shape function spaces of nine-node triangular plate elements converging for any meshes, which generalize Spect's element and Veubeke's element.
A reduced-order dynamic model for an unbalanced rotor system is developed, taking the coupling between torsional and lateral vibrations into account. It is assumed that a shaft is regarded as a continuous viscoelastic...A reduced-order dynamic model for an unbalanced rotor system is developed, taking the coupling between torsional and lateral vibrations into account. It is assumed that a shaft is regarded as a continuous viscoelastic shaft with unbalanced and small deformation properties. The equations of motion for the torsional and lateral vibrations are derived using Lagrange's approach with the frequency-dependent shape function. The rotor torsional vibration is coupled with the lateral vibrations by unbalance elements in a way of excitations. Simulation and experiment results show clearly that the torsional vibration has strong impact on the rotor lateral vibrations, and it causes subharmonic and superharmonic excitations through unbalance elements, which leads to the superharmonic resonances in the lateral vibrations. This model with low-order and high accuracy is suitable for rotor dynamic analysis in real time simulation as well as for active vibration control syntheses.展开更多
Lightweight porous materials with high load-bearing,damage tolerance and energy absorption(EA)as well as intelligence of shape recovery after material deformation are beneficial and critical for many applications,e.g....Lightweight porous materials with high load-bearing,damage tolerance and energy absorption(EA)as well as intelligence of shape recovery after material deformation are beneficial and critical for many applications,e.g.aerospace,automobiles,electronics,etc.Cuttlebone produced in the cuttlefish has evolved vertical walls with the optimal corrugation gradient,enabling stress homogenization,significant load bearing,and damage tolerance to protect the organism from high external pressures in the deep sea.This work illustrated that the complex hybrid wave shape in cuttlebone walls,becoming more tortuous from bottom to top,creates a lightweight,load-bearing structure with progressive failure.By mimicking the cuttlebone,a novel bionic hybrid structure(BHS)was proposed,and as a comparison,a regular corrugated structure and a straight wall structure were designed.Three types of designed structures have been successfully manufactured by laser powder bed fusion(LPBF)with NiTi powder.The LPBF-processed BHS exhibited a total porosity of 0.042% and a good dimensional accuracy with a peak deviation of 17.4μm.Microstructural analysis indicated that the LPBF-processed BHS had a strong(001)crystallographic orientation and an average size of 9.85μm.Mechanical analysis revealed the LPBF-processed BHS could withstand over 25000 times its weight without significant deformation and had the highest specific EA value(5.32 J·g^(−1))due to the absence of stress concentration and progressive wall failure during compression.Cyclic compression testing showed that LPBF-processed BHS possessed superior viscoelastic and elasticity energy dissipation capacity.Importantly,the uniform reversible phase transition from martensite to austenite in the walls enables the structure to largely recover its pre-deformation shape when heated(over 99% recovery rate).These design strategies can serve as valuable references for the development of intelligent components that possess high mechanical efficiency and shape memory capabilities.展开更多
An interesting discretization method for Helmholtz equations was introduced in B.Despres[1].This method is based on the ultra weak variational formulation(UWVF)and the wave shape functions,which are exact solutions of...An interesting discretization method for Helmholtz equations was introduced in B.Despres[1].This method is based on the ultra weak variational formulation(UWVF)and the wave shape functions,which are exact solutions of the governing Helmholtz equation.In this paper we are concerned with fast solver for the system generated by the method in[1].We propose a new preconditioner for such system,which can be viewed as a combination between a coarse solver and the block diagonal preconditioner introduced in[13].In our numerical experiments,this preconditioner is applied to solve both two-dimensional and three-dimensional Helmholtz equations,and the numerical results illustrate that the new preconditioner is much more efficient than the original block diagonal preconditioner.展开更多
A novel strategy of probability density function （PDF） shape control is proposed in stochastic systems. The control er is designed whose parameters are optimal y obtained through the improved particle swarm optimiza...A novel strategy of probability density function （PDF） shape control is proposed in stochastic systems. The control er is designed whose parameters are optimal y obtained through the improved particle swarm optimization algorithm. The parameters of the control er are viewed as the space position of a particle in particle swarm optimization algorithm and updated continual y until the control er makes the PDF of the state variable as close as possible to the expected PDF. The proposed PDF shape control technique is compared with the equivalent linearization technique through simulation experiments. The results show the superiority and the effectiveness of the proposed method. The control er is excellent in making the state PDF fol ow the expected PDF and has the very smal error between the state PDF and the expected PDF, solving the control problem of the PDF shape in stochastic systems effectively.展开更多
A shape hardening function is developed that improves the predictive capabilities of the generalized bounding surface model for cohesive soils, especially when applied to overconsolidated specimens. This improvement i...A shape hardening function is developed that improves the predictive capabilities of the generalized bounding surface model for cohesive soils, especially when applied to overconsolidated specimens. This improvement is realized without any changes to the simple elliptical shape of the bounding surface, and actually reduces the number of parameters associated with the model by one.展开更多
The shape control of probability density function(PDF) of the system state is an important topic in stochastic systems. In this paper, we propose a control technique for PDF shape of the state variable in nonlinear st...The shape control of probability density function(PDF) of the system state is an important topic in stochastic systems. In this paper, we propose a control technique for PDF shape of the state variable in nonlinear stochastic systems. Firstly, we derive and prove the form of the controller by investigating the Fokker-PlanckKolmogorov(FPK) equation arising from the stochastic system. Secondly, an approach for getting approximate solution of the FPK equation is provided. A special function including some parameters is taken as the approximate stationary solution of the FPK equation. We use nonlinear least square method to solve the parameters in the function, and capture the approximate solution of the FPK equation. Substituting the approximate solution into the form of the controller, we can acquire the PDF shape controller. Lastly, some example simulations are conducted to verify the algorithm.展开更多
Based on the isomorphism between the space of star-shaped sets and the space of continuous positively homogeneous real-valued functions, the star-shaped differential of a directionally differentiable function is defin...Based on the isomorphism between the space of star-shaped sets and the space of continuous positively homogeneous real-valued functions, the star-shaped differential of a directionally differentiable function is defined. Formulas for star-shaped differential of a pointwise maximum and a pointwise minimum of a finite number of directionally differentiable functions, and a composite of two directionaUy differentiable functions are derived. Furthermore, the mean-value theorem for a directionaUy differentiable function is demonstrated.展开更多
The torsional impact response of a penny-shaped crack in a nonhomogeneous strip is considered. The shear modulus is assumed to be functionally graded such that the mathematics is tractable. Laplace and Hankel transfor...The torsional impact response of a penny-shaped crack in a nonhomogeneous strip is considered. The shear modulus is assumed to be functionally graded such that the mathematics is tractable. Laplace and Hankel transforms were used to reduce the problem to solving a Fredholm integral equation. The crack tip stress field is obtained by considering the asymptotic behavior of Bessel function. Explicit expressions of both the dynamic stress intensity factor and the energy density factor were derived.And it is shown that, as crack driving force, they are equivalent for the present crack problem. Investigated are the effects of material nonhomogeneity and (strip's) highness on the dynamic fracture behavior. Numerical results reveal that the peak of the dynamic stress intensity factor can be suppressed by increasing the nonhomogeneity parameter of the shear modulus, and that the dynamic behavior varies little with the adjusting of the strip's highness.展开更多
We establish the concept of shapes of functions by using partial differential inequalites. Our definition about shapes includes some usual shapes such as convex,subharmonic,etc.,and gives many new shapes of functions....We establish the concept of shapes of functions by using partial differential inequalites. Our definition about shapes includes some usual shapes such as convex,subharmonic,etc.,and gives many new shapes of functions.The main results show that the shape preserving approxi- mation has close relation to the shape preserving extension.One of our main results shows that if f∈C(Ω)has some shape defined by our definition,then f can be uniformly approximated by polynomials P_n ∈p_n(n∈N)which have the same shape in Ω,and the degree of the ap- proximation is Cω(f,n^(-β))with constants C,β>0.展开更多
A nonlinear finite element method is applied to observe how inclusion shape influence the thermal response of a ceramic-metal functionally graded material (FGM). The elastic and plastic behaviors of the layers which a...A nonlinear finite element method is applied to observe how inclusion shape influence the thermal response of a ceramic-metal functionally graded material (FGM). The elastic and plastic behaviors of the layers which are two-phase isotropic composites consisting of randomly oriented elastic spheroidal Inclusions and a ductile matrix are predicted by cc mean field method. The prediction results show that inclusion shape has remarkable influence on the overall behavior of the composite. The consequences of the thermal response analysis of the FGM are that the response is dependent on inclusion shape and its composition profile cooperatively and that the plastic behavior of each layer should be taken into account in optimum design of a ceramic-metal FGM.展开更多
To assess the left ventricle regional systolic and diastolic function, left ventricle geometry and left venticle sphericity indexes in patients with dilated cardiomyopathy （DCM） by quantitative tissue velocity imagi...To assess the left ventricle regional systolic and diastolic function, left ventricle geometry and left venticle sphericity indexes in patients with dilated cardiomyopathy （DCM） by quantitative tissue velocity imaging （QTVI）. Methods Thirty normal subjects and 52 DCM patients underwent QTVI and colour Doppler flow imaging study in order to measure the left ventricular regional function along left ventricle apical long-axis view and the left ventricle geometry. Peak tissue velocities of left venticle regional muscular tissue during systole （ Vs）, systolic acceleration （ a）, early diastole（Ve） and left atrium contraction（Va） along left venticle apical long axis view were measured. The indexes of left ventricular regional systolic and diastolic function were mearsured at the same time. The left ventricle geometry shape was reflected from the systolic and diastolic sphericity index （Sis and Sid）, the left ventricular ejection fraction （LVEF） and D wave / A wave （PVd/PVa） of pulmonary veins flowing spectrum reflected the global left ventricular systolic and diastolic function. The Vs, Ve, Va, a, PVd/PVa ratio, LVEF, Sis, Sid and their correlations between normal subjects and patients with DCM were compared and analyzed. Results Vs, Ve, Va, a, PVd/PVa, Sis and Sid in patients with DCM were lower than those in normal persons. There were significant relations between Sis and a （ r = 0. 6142, P 〈 0. 05）, Ve/Va and Sid （ r = 0. 6271, P 〈 0. 05 ）. Conclusions QTVI offer a newer method which has a higher sensitivity and accuracy in evaluating the left venticle regional systolic and diastolic function in DCM patients. There was significant relation between regional cardiac function and left venticle sphericity. （ S Chin J Cardiol 2009; 10（1） ： 9 -14）展开更多
The penalty function method of continuum shape optimization and its sensitivity analysis technique are presented. A relatively simple integrated shape optimization system is developed and used to optimize the design o...The penalty function method of continuum shape optimization and its sensitivity analysis technique are presented. A relatively simple integrated shape optimization system is developed and used to optimize the design of the inner frame shape of a three-axis test table. The result shows that the method converges well, and the system is stable and reliable.展开更多
The paper introduces a method to get three-dimensional reproduction of log shape by adopting Spline Function in fitting the curve of the finite log data. The method has ad-vantages of higher accuracy, less acquired da...The paper introduces a method to get three-dimensional reproduction of log shape by adopting Spline Function in fitting the curve of the finite log data. The method has ad-vantages of higher accuracy, less acquired data, easier to use, etc. Making use of high-precision drawing function of computer, the graphs of log geometric shape in different visual angles can be achieved easily with this method. It also provided a firm foundation for the determination of optimum saw cutting scheme.展开更多
文摘In the optimal control problem of nonlinear dynamical system,the Hamiltonian formulation is useful and powerful to solve an optimal control force.However,the resulting Euler-Lagrange equations are not easy to solve,when the performance index is complicated,because one may encounter a two-point boundary value problem of nonlinear differential algebraic equations.To be a numerical method,it is hard to exactly preserve all the specified conditions,which might deteriorate the accuracy of numerical solution.With this in mind,we develop a novel algorithm to find the solution of the optimal control problem of nonlinear Duffing oscillator,which can exactly satisfy all the required conditions for the minimality of the performance index.A new idea of shape functions method(SFM)is introduced,from which we can transform the optimal control problems to the initial value problems for the new variables,whose initial values are given arbitrarily,and meanwhile the terminal values are determined iteratively.Numerical examples confirm the high-performance of the iterative algorithms based on the SFM,which are convergence fast,and also provide very accurate solutions.The new algorithm is robust,even large noise is imposed on the input data.
文摘Many mechanical problems can be induced from differential equations with boundary conditions; there exist analytic and numerical methods for solving the differential equations. Usually it is not so easy to obtain analytic solutions. So it is necessary to give numerical solutions. The reproducing kernel particle （RKP） method is based on the Carlerkin Meshless method. According to the Sobolev space and Fourier transform, the RKP shape function is mathematically proved in this paper.
文摘Using the method of undetermined coefficients, we construct a set of shape function spaces of nine-node triangular plate elements converging for any meshes, which generalize Spect's element and Veubeke's element.
基金Project(51105017)supported by National Natural Science Foundation of ChinaProject(2011BAG09B00)supported by the National Science and Technology Support Program,ChinaProject(2010DFB80020)supported by the National Science and Technology Major Project of the Ministry of Science and Technology of China
文摘A reduced-order dynamic model for an unbalanced rotor system is developed, taking the coupling between torsional and lateral vibrations into account. It is assumed that a shaft is regarded as a continuous viscoelastic shaft with unbalanced and small deformation properties. The equations of motion for the torsional and lateral vibrations are derived using Lagrange's approach with the frequency-dependent shape function. The rotor torsional vibration is coupled with the lateral vibrations by unbalance elements in a way of excitations. Simulation and experiment results show clearly that the torsional vibration has strong impact on the rotor lateral vibrations, and it causes subharmonic and superharmonic excitations through unbalance elements, which leads to the superharmonic resonances in the lateral vibrations. This model with low-order and high accuracy is suitable for rotor dynamic analysis in real time simulation as well as for active vibration control syntheses.
基金supported by the National Natural Science Foundation of China(Grant No.52225503)National Key Research and Development Program of China(Grant No.2022YFB3805701)+1 种基金Development Program of Jiangsu Province(Grant Nos.BE2022069 and BE2022069-1)Postgraduate Research&Practice Innovation Program of Jiangsu Province(Grant No.KYCX21-0207).
文摘Lightweight porous materials with high load-bearing,damage tolerance and energy absorption(EA)as well as intelligence of shape recovery after material deformation are beneficial and critical for many applications,e.g.aerospace,automobiles,electronics,etc.Cuttlebone produced in the cuttlefish has evolved vertical walls with the optimal corrugation gradient,enabling stress homogenization,significant load bearing,and damage tolerance to protect the organism from high external pressures in the deep sea.This work illustrated that the complex hybrid wave shape in cuttlebone walls,becoming more tortuous from bottom to top,creates a lightweight,load-bearing structure with progressive failure.By mimicking the cuttlebone,a novel bionic hybrid structure(BHS)was proposed,and as a comparison,a regular corrugated structure and a straight wall structure were designed.Three types of designed structures have been successfully manufactured by laser powder bed fusion(LPBF)with NiTi powder.The LPBF-processed BHS exhibited a total porosity of 0.042% and a good dimensional accuracy with a peak deviation of 17.4μm.Microstructural analysis indicated that the LPBF-processed BHS had a strong(001)crystallographic orientation and an average size of 9.85μm.Mechanical analysis revealed the LPBF-processed BHS could withstand over 25000 times its weight without significant deformation and had the highest specific EA value(5.32 J·g^(−1))due to the absence of stress concentration and progressive wall failure during compression.Cyclic compression testing showed that LPBF-processed BHS possessed superior viscoelastic and elasticity energy dissipation capacity.Importantly,the uniform reversible phase transition from martensite to austenite in the walls enables the structure to largely recover its pre-deformation shape when heated(over 99% recovery rate).These design strategies can serve as valuable references for the development of intelligent components that possess high mechanical efficiency and shape memory capabilities.
基金The second author was supported by the Major Research Plan of Natural Science Foundation of China G91130015the Key Project of Natural Science Foundation of China G11031006National Basic Research Program of China G2011309702.
文摘An interesting discretization method for Helmholtz equations was introduced in B.Despres[1].This method is based on the ultra weak variational formulation(UWVF)and the wave shape functions,which are exact solutions of the governing Helmholtz equation.In this paper we are concerned with fast solver for the system generated by the method in[1].We propose a new preconditioner for such system,which can be viewed as a combination between a coarse solver and the block diagonal preconditioner introduced in[13].In our numerical experiments,this preconditioner is applied to solve both two-dimensional and three-dimensional Helmholtz equations,and the numerical results illustrate that the new preconditioner is much more efficient than the original block diagonal preconditioner.
基金supported by the National Natural Science Fundation of China(61273127)the Specialized Research Fund of the Doctoral Program in Higher Education(20106118110009+2 种基金20116118110008)the Scientific Research Plan Projects of Shaanxi Education Department(12JK0524)the Young Teachers Scientific Research Fund of Xi’an University of Posts and Telecommunications(1100434)
文摘A novel strategy of probability density function （PDF） shape control is proposed in stochastic systems. The control er is designed whose parameters are optimal y obtained through the improved particle swarm optimization algorithm. The parameters of the control er are viewed as the space position of a particle in particle swarm optimization algorithm and updated continual y until the control er makes the PDF of the state variable as close as possible to the expected PDF. The proposed PDF shape control technique is compared with the equivalent linearization technique through simulation experiments. The results show the superiority and the effectiveness of the proposed method. The control er is excellent in making the state PDF fol ow the expected PDF and has the very smal error between the state PDF and the expected PDF, solving the control problem of the PDF shape in stochastic systems effectively.
基金supported by the Fulbright Colombia-Colciencias Scholarship and Universidad Militar Nueva Granada
文摘A shape hardening function is developed that improves the predictive capabilities of the generalized bounding surface model for cohesive soils, especially when applied to overconsolidated specimens. This improvement is realized without any changes to the simple elliptical shape of the bounding surface, and actually reduces the number of parameters associated with the model by one.
基金the National Natural Science Foundation of China(No.61273127)the Specialized Research Fund for the Doctoral Program of Higher Education of China(No.20116118110008)the Scientific Research Plan Projects of Shaanxi Education Department(No.12JK0524)
文摘The shape control of probability density function(PDF) of the system state is an important topic in stochastic systems. In this paper, we propose a control technique for PDF shape of the state variable in nonlinear stochastic systems. Firstly, we derive and prove the form of the controller by investigating the Fokker-PlanckKolmogorov(FPK) equation arising from the stochastic system. Secondly, an approach for getting approximate solution of the FPK equation is provided. A special function including some parameters is taken as the approximate stationary solution of the FPK equation. We use nonlinear least square method to solve the parameters in the function, and capture the approximate solution of the FPK equation. Substituting the approximate solution into the form of the controller, we can acquire the PDF shape controller. Lastly, some example simulations are conducted to verify the algorithm.
文摘Based on the isomorphism between the space of star-shaped sets and the space of continuous positively homogeneous real-valued functions, the star-shaped differential of a directionally differentiable function is defined. Formulas for star-shaped differential of a pointwise maximum and a pointwise minimum of a finite number of directionally differentiable functions, and a composite of two directionaUy differentiable functions are derived. Furthermore, the mean-value theorem for a directionaUy differentiable function is demonstrated.
文摘The torsional impact response of a penny-shaped crack in a nonhomogeneous strip is considered. The shear modulus is assumed to be functionally graded such that the mathematics is tractable. Laplace and Hankel transforms were used to reduce the problem to solving a Fredholm integral equation. The crack tip stress field is obtained by considering the asymptotic behavior of Bessel function. Explicit expressions of both the dynamic stress intensity factor and the energy density factor were derived.And it is shown that, as crack driving force, they are equivalent for the present crack problem. Investigated are the effects of material nonhomogeneity and (strip's) highness on the dynamic fracture behavior. Numerical results reveal that the peak of the dynamic stress intensity factor can be suppressed by increasing the nonhomogeneity parameter of the shear modulus, and that the dynamic behavior varies little with the adjusting of the strip's highness.
文摘We establish the concept of shapes of functions by using partial differential inequalites. Our definition about shapes includes some usual shapes such as convex,subharmonic,etc.,and gives many new shapes of functions.The main results show that the shape preserving approxi- mation has close relation to the shape preserving extension.One of our main results shows that if f∈C(Ω)has some shape defined by our definition,then f can be uniformly approximated by polynomials P_n ∈p_n(n∈N)which have the same shape in Ω,and the degree of the ap- proximation is Cω(f,n^(-β))with constants C,β>0.
基金Funded by National Science Foundation of China(Grant:1987205).
文摘A nonlinear finite element method is applied to observe how inclusion shape influence the thermal response of a ceramic-metal functionally graded material (FGM). The elastic and plastic behaviors of the layers which are two-phase isotropic composites consisting of randomly oriented elastic spheroidal Inclusions and a ductile matrix are predicted by cc mean field method. The prediction results show that inclusion shape has remarkable influence on the overall behavior of the composite. The consequences of the thermal response analysis of the FGM are that the response is dependent on inclusion shape and its composition profile cooperatively and that the plastic behavior of each layer should be taken into account in optimum design of a ceramic-metal FGM.
文摘To assess the left ventricle regional systolic and diastolic function, left ventricle geometry and left venticle sphericity indexes in patients with dilated cardiomyopathy （DCM） by quantitative tissue velocity imaging （QTVI）. Methods Thirty normal subjects and 52 DCM patients underwent QTVI and colour Doppler flow imaging study in order to measure the left ventricular regional function along left ventricle apical long-axis view and the left ventricle geometry. Peak tissue velocities of left venticle regional muscular tissue during systole （ Vs）, systolic acceleration （ a）, early diastole（Ve） and left atrium contraction（Va） along left venticle apical long axis view were measured. The indexes of left ventricular regional systolic and diastolic function were mearsured at the same time. The left ventricle geometry shape was reflected from the systolic and diastolic sphericity index （Sis and Sid）, the left ventricular ejection fraction （LVEF） and D wave / A wave （PVd/PVa） of pulmonary veins flowing spectrum reflected the global left ventricular systolic and diastolic function. The Vs, Ve, Va, a, PVd/PVa ratio, LVEF, Sis, Sid and their correlations between normal subjects and patients with DCM were compared and analyzed. Results Vs, Ve, Va, a, PVd/PVa, Sis and Sid in patients with DCM were lower than those in normal persons. There were significant relations between Sis and a （ r = 0. 6142, P 〈 0. 05）, Ve/Va and Sid （ r = 0. 6271, P 〈 0. 05 ）. Conclusions QTVI offer a newer method which has a higher sensitivity and accuracy in evaluating the left venticle regional systolic and diastolic function in DCM patients. There was significant relation between regional cardiac function and left venticle sphericity. （ S Chin J Cardiol 2009; 10（1） ： 9 -14）
文摘The penalty function method of continuum shape optimization and its sensitivity analysis technique are presented. A relatively simple integrated shape optimization system is developed and used to optimize the design of the inner frame shape of a three-axis test table. The result shows that the method converges well, and the system is stable and reliable.
文摘The paper introduces a method to get three-dimensional reproduction of log shape by adopting Spline Function in fitting the curve of the finite log data. The method has ad-vantages of higher accuracy, less acquired data, easier to use, etc. Making use of high-precision drawing function of computer, the graphs of log geometric shape in different visual angles can be achieved easily with this method. It also provided a firm foundation for the determination of optimum saw cutting scheme.
