带接触力的欧拉-伯努利梁：存在性、唯一性与数值解
Euler-Bernoulli beams with contact forces: existence, uniqueness, and numerical solutions
AI总结：
本文研究带接触力的欧拉-伯努利四阶边值问题，证明了解的存在唯一性，并通过有限差分离散和不动点迭代给出数值求解方法，证明了离散解收敛到连续解。
摘要：
本文研究欧拉-伯努利四阶边值问题（BVP）$w^{(4)}=f(x,w)$，$x\in \intcc{a,b}$，并在端点给定 $w$ 和 $w''$ 的值，其中右端项 $f$ 的行为受生物力学、机电和结构应用中包含接触力的情形所驱动。特别地，我们考虑 $f$ 关于其第二个自变量有上界且单调递减的情况。首先，我们证明该边值问题解的存在性和唯一性。随后，我们研究该边值问题的数值解，采用有限差分进行空间离散。与原始连续空间问题类似，离散问题也始终具有唯一解。在 $f$ 为分段线性实例的情形下，离散问题构成绝对值方程的一个例子。我们证明该绝对值方程的解可以通过不动点迭代获得，并且绝对值方程的解收敛到连续边值问题的解。我们还通过一个数值例子展示了不动点迭代的性能。
英文摘要：
In this paper, we investigate the Euler-Bernoulli fourth-order boundary value problem (BVP) $w^{(4)}=f(x,w)$, $x\in \intcc{a,b}$, with specified values of $w$ and $w''$ at the end points, where the behaviour of the right-hand side $f$ is motivated by biomechanical, electromechanical, and structural applications incorporating contact forces. In particular, we consider the case when $f$ is bounded above and monotonically decreasing with respect to its second argument. First, we prove the existence and uniqueness of solutions to the BVP. We then study numerical solutions to the BVP, where we resort to spatial discretization by means of finite difference. Similar to the original continuous-space problem, the discrete problem always possesses a unique solution. In the case of a piecewise linear instance of $f$, the discrete problem is an example of the absolute value equation. We show that solutions to this absolute value equation can be obtained by means of fixed-point iterations, and that solutions to the absolute value equation converge to solutions of the continuous BVP. We also illustrate the performance of the fixed-point iterations through a numerical example.
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