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This multigrid solver is proven to provide the same accuracy as the finite element solution with respect to energy norm, and the computational complexity is suboptimal.

并证明用共轭梯度法作迭代算子时，该方法可以达到能量模意义下的最优，计算复杂度接近最优。

So three different method, in which, under the control of the energy norm error, the orders of the shape functions in the whole region, in some elements, and on some generalized nodes are increased respectively, are adopted to implement the p-version adaptive hierarchical Block Element Method.

文中采用整体升阶、单元升阶和广义结点升阶三种方法，在能量范数误差的控制下分别升高所有单元、部分单元和部分广义结点的形函数阶数，从而实现了岩体的p型自适应阶谱块体单元法。

By using the modal superposition method for the dynamic responses calculation of the linear oscillation system, a judging indictor in the energy norm form is given to determine the number of dominant modes on mode truncation .

针对用模态叠加法计算结构线性振动系统的响应，作模态截断的截尾模态阶数的取值，给出了能量范数形式的判断指标。

By using the modal superposition method for the dynamic responses calculation of the linear oscillation system, a judging indictor in the energy norm form is given to determine the number of dominant modes on mode truncation.

针对用模态叠加法计算结构线性振动系统的响应，作模态截断的截尾模态阶数的取值，给出了能量范数形式的判断指标。用于小湾拱坝结构的动力分析表明，需取 10阶以上的模态进行叠加

By re-constructing the system, the residual error dynamic characteristics of unknown input and fault signals including unknown disturbances and modelling error are obtained. The energy norm indexes of disturbance and fault signals of residual error are selected separately to reflect the restraint of disturbance and the sensitivity of faults, and the design of fault detection observer is described as an optimization problem.

通过对系统进行重构，获取了包括未知扰动、建模误差等未知输入信号和故障信号的残差动态特性，分别选取残差的扰动信号和故障信号的能量范数指标来体现其对扰动的抑制作用和对故障的灵敏性，并将故障检测观测器的设计描述为一个优化问题。

Since the modes of foundation are dense,by taking the criterion of energy response as mode truncation index,the energy norm and criterion are established to calculate the mode truncated order of dynamic compaction response.

由于地基模态密集，提出以能量响应信息的收敛作为模态截断的判断指标，建立能量范数和能量判据，计算强夯动力响应的模态截断阶数。

In the framework of non-equilibrium elastoplastic mechanics, the classic finite element method is extended to the analysis for post-failure state of structure, where the plastic complementary energy norm serves as the criterion of global stability while the unbalanced force and its development indicates the pattern and path of destruction.

在非平衡态弹塑性力学框架下，将经典有限元拓展到结构失稳后的分析，塑性余能范数即可作为库群失稳判据，而不平衡力及其发展则表征了库群失稳破坏的形态和演化路径。

The model as follows:〓 u =0,∈〓Ω×J u =u〓, x∈Ω chapter 3 is devoted to study the moving grid timediscretization FEM of above model, the optimal energy norm error estimate is obtained.

本文第二章重点研究了上述模型有限元算法的构造及收敛性，克服了ρ同时依赖压力p、浓度c带来的困难，证明了有限元半离散解的整体存在性和最优误差估计。

The adaptive program is calculated by T-B stress recovery scheme and z-z error estimation, the error is estimated by energy norm of difference of recovery stress and meshless stress. If the local relative error exceeds the limit relative error, then new nodes will be generated at the high error area.

本研究的自适应方法采用TB应力平滑技术和Z-Z误差估计，利用平滑应力和无网格法所得应力解差值的能量范数来估算误差，如果局部相对误差超过了允许误差，则在高误差域插入新的离散点进行加密。

A high accuracy finite volume element method was given based on cubic spline interpolation for linear and nonlinear twopoint boundary value problems of ordinary differential equations. The computational scheme was derived and the property of positive type was discussed. Further we presented the convergence analysis and the scheme was proved having fourth order error estimate in discrete energy norm.

针对常微分方程线性和非线性两点边值问题，提出了基于三次样条插值的高精度有限体积元方法，给出了具体计算格式，讨论了格式所具有的优良性质--正型性，并应用能量方法给出了收敛性分析，证明了格式按照离散能量模具有四阶精度。

The network space is the typical form of abstract space when it comes to the digital stage.

网络空间是抽象空间数字化阶段的典型形式。

The Notarial System,as the system of judicial certification in our country,is both the constructor and the defender of social credit system,It has its special value and function in the social honesty.

公证制度作为我国司法证明制度，既是社会信用体系的建设者，同时也是社会信用的捍卫者，在诚信中有其特殊的价值和功能。